https://peda.net/id/c5254890e26 2018-12-13T21:31:28+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/ba5808e8ff0 2018-12-13T21:31:46+02:00 <span>446</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Ccos2x%2B2%5Csin%20x" alt="f\left(x\right)=\cos2x+2\sin x"/></div> <div>Sinifunktion sinx perujakso on 2π ja cos2x perusjakso on 2π/2=π. Funktion cos2 arvot toistuvat samoina π:n välein</div> <div>ja myös 2π:n välein. Siis funktioniden sinix ja cos2x arvot toistuvat samoina 2π:n välein eli funktion f(x) jakso on 2π. </div> <div> </div> <div>Riittää etsiä funktion f(x) suurin ja pienin arvo välillä [0,2π]. </div> <div>Suljetulla välillä jatkuva funktio saa suurimman ja pienimmän arvonsa välin pätepisteisä tai välillä olevissa derivaatan nollakohdissa.</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D-2%5Csin2x%2B2%5Ccos%20x" alt="f'\left(x\right)=-2\sin2x+2\cos x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D0" alt="f'\left(x\right)=0"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Csin2x%2B2%5Ccos%20x%3D0" alt="-2\sin2x+2\cos x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Ccdot2%5Csin%20x%5Ccos%20x%2B2%5Ccos%20x%3D0" alt="-2\cdot2\sin x\cos x+2\cos x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%5Cleft(-2%5Csin%20x%2B1%5Cright)%3D0" alt="2\cos x\left(-2\sin x+1\right)=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%3D0" alt="2\cos x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D0" alt="\cos x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi%5C%20%5C%20" alt="x=\frac{\pi}{2}+n\cdot\pi\ \ "/> <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Csin%20x%2B1%3D0" alt="-2\sin x+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Csin%20x%3D-1" alt="-2\sin x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin x=\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{\pi}{6}+n\cdot2\pi"/><br/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\pi-\frac{\pi}{6}+n\cdot2\pi=\frac{5\pi}{6}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/></div> <div> </div> <div>Välillä [0,2π] ovat nollatkohdat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2B%5Cpi%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%7B%2C%7D%5C%20x%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="x=\frac{\pi}{2}{,}\ x=\frac{\pi}{2}+\pi=\frac{3\pi}{2}{,}\ x=\frac{\pi}{6}{,}\ x=\frac{5\pi}{6}"/></div> <div>Lasketaan funktion arvot välin päätepisteissä ja välillä olevissa derivaatan nollakohdissa</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D%5Ccos%5Cleft(2%5Ccdot0%5Cright)%2B2%5Csin0%3D1%2B2%5Ccdot0%3D1" alt="f\left(0\right)=\cos\left(2\cdot0\right)+2\sin0=1+2\cdot0=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(2%5Cpi%5Cright)%3D%5Ccos%5Cleft(2%5Ccdot2%5Cpi%5Cright)%2B2%5Csin%5Cleft(2%5Cpi%5Cright)%3D1%2B2%5Ccdot0%3D1" alt="f\left(2\pi\right)=\cos\left(2\cdot2\pi\right)+2\sin\left(2\pi\right)=1+2\cdot0=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D1" alt="f\left(\frac{\pi}{2}\right)=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B3%5Cpi%7D%7B2%7D%5Cright)%3D-3" alt="f\left(\frac{3\pi}{2}\right)=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D%5Cfrac%7B3%7D%7B2%7D" alt="f\left(\frac{\pi}{6}\right)=\frac{3}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B5%5Cpi%7D%7B6%7D%5Cright)%3D%5Cfrac%7B3%7D%7B2%7D" alt="f\left(\frac{5\pi}{6}\right)=\frac{3}{2}"/></div> <div>Arvojoukko om [-3,3/2]</div> <div><br/> <div> </div> </div> 2018-12-13T21:31:46+02:00 https://peda.net/id/50983f8cfde 2018-12-12T10:09:57+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=Du%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%27%5Cleft(s%5Cleft(x%5Cright)%5Cright)%5Ccdot%20s%27%5Cleft(x%5Cright)" alt="Du\left(s\left(x\right)\right)=u'\left(s\left(x\right)\right)\cdot s'\left(x\right)"/><br/> <div>Esim. Derivoi</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft(2x%2B4%5Cright)%5E5" alt="f\left(x\right)=\left(2x+4\right)^5"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D2x%2B4" alt="s\left(x\right)=2x+4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E5" alt="u\left(x\right)=x^5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%27%5Cleft(x%5Cright)%3D2%7B%2C%7D%5C%20u%27%5Cleft(x%5Cright)%3D5x%5E4" alt="s'\left(x\right)=2{,}\ u'\left(x\right)=5x^4"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%27%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D5x%5Cleft(2x%2B4%5Cright)%5E4%5Ccdot2%3D10x%5Cleft(2x%2B4%5Cright)%5E4" alt="u'\left(s\left(x\right)\right)=5x\left(2x+4\right)^4\cdot2=10x\left(2x+4\right)^4"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%5Cleft(x%5Cright)%3D2x%5Ccdot%5Ctan%5Cleft(3x%5Cright)" alt="p\left(x\right)=2x\cdot\tan\left(3x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=Df%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%3Df%27%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%2Bg%27%5Cleft(x%5Cright)f%5Cleft(x%5Cright)" alt="Df\left(x\right)g\left(x\right)=f'\left(x\right)g\left(x\right)+g'\left(x\right)f\left(x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2x" alt="f\left(x\right)=2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Ctan%5Cleft(3x%5Cright)" alt="g\left(x\right)=\tan\left(3x\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%27%5Cleft(x%5Cright)%3D3" alt="s'\left(x\right)=3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%27%5Cleft(x%5Cright)%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D%5C%20tai%5C%20u%27%5Cleft(x%5Cright)%3D1%2B%5Ctan%5E2x" alt="u'\left(x\right)=\frac{1}{\cos^2x}\ tai\ u'\left(x\right)=1+\tan^2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%27%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7B%5Ccos%5E2%5Cleft(3x%5Cright)%7D%5C%20%5C%20tai%5C%20g%27%5Cleft(x%5Cright)%3D3%2B3%5Ctan%5E2%5Cleft(3x%5Cright)" alt="g'\left(x\right)=\frac{3}{\cos^2\left(3x\right)}\ \ tai\ g'\left(x\right)=3+3\tan^2\left(3x\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=p%27%5Cleft(x%5Cright)%3D2%5Ccdot%5Ctan%5Cleft(3x%5Cright)%2B2x%5Ccdot%5Cleft(3%2B3%5Ctan%5E2%5Cleft(3x%5Cright)%5Cright)%3D2%5Ctan%5Cleft(3x%5Cright)%2B6x%2B6x%5Ctan%5E2%5Cleft(3x%5Cright)" alt="p'\left(x\right)=2\cdot\tan\left(3x\right)+2x\cdot\left(3+3\tan^2\left(3x\right)\right)=2\tan\left(3x\right)+6x+6x\tan^2\left(3x\right)"/> </div> </div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Cleft(3x%5E2%2B2x%5Cright)%5E3%7D%7B2x%7D" alt="r\left(x\right)=\frac{\left(3x^2+2x\right)^3}{2x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bg%5Cleft(x%5Cright)%7D%3D%5Cfrac%7Bf%27%5Cleft(x%5Cright)%5Ccdot%20g%5Cleft(x%5Cright)-g%27%5Cleft(x%5Cright)%5Ccdot%20f%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E2%7D" alt="D\frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)\cdot g\left(x\right)-g'\left(x\right)\cdot f\left(x\right)}{\left(g\left(x\right)\right)^2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3D3%5Cleft(3x%5E2%2B2x%5Cright)%5E2%5Cleft(6x%2B2%5Cright)" alt="f'\left(x\right)=3\left(3x^2+2x\right)^2\left(6x+2\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=r%27%5Cleft(x%5Cright)%3D%5Cfrac%7B3%5Cleft(3x%5E2%2B2x%5Cright)%5E2%5Cleft(6x%2B2%5Cright)%5Ccdot2x-2%5Ccdot%5Cleft(3x%5E2%2B2x%5Cright)%5E3%7D%7B%5Cleft(2x%5Cright)%5E2%7D%3D%5Cfrac%7B2%5Cleft(3x%5E2%2B2x%5Cright)%5E2%5Cleft(3x%5Cleft(6x%2B2%5Cright)-%5Cleft(3x%5E2%2B2x%5Cright)%5Cright)%7D%7B4x%5E2%7D%3D%5Cfrac%7B%5Cleft(3x%5E2%2B2x%5Cright)%5E2%5Cleft(18x%5E2%2B6x-3x%5E2-2x%5Cright)%7D%7B2x%5E2%7D%3D%5Cfrac%7B%5Cleft(3x%5E2%2B2x%5Cright)%5E2%5Cleft(15x%5E2%2B4x%5Cright)%7D%7B2x%5E2%7D" alt="r'\left(x\right)=\frac{3\left(3x^2+2x\right)^2\left(6x+2\right)\cdot2x-2\cdot\left(3x^2+2x\right)^3}{\left(2x\right)^2}=\frac{2\left(3x^2+2x\right)^2\left(3x\left(6x+2\right)-\left(3x^2+2x\right)\right)}{4x^2}=\frac{\left(3x^2+2x\right)^2\left(18x^2+6x-3x^2-2x\right)}{2x^2}=\frac{\left(3x^2+2x\right)^2\left(15x^2+4x\right)}{2x^2}"/> 2018-12-12T10:09:57+02:00 https://peda.net/id/c9cbcf94f7b 2018-12-04T13:50:25+02:00 <span>Määritelmä</span> <div>Lauseke<img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)" alt="u\left(s\left(x\right)\right)"/>on funktioiden <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)" alt="u\left(x\right)"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)" alt="s\left(x\right)"/>yhdistetty funktio</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)" alt="u\left(x\right)"/>on ulkofunktio ja <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)" alt="s\left(x\right)"/>on sisäfunktio</div> <div>Määtitään<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%3Du%5Cleft(s%5Cleft(x%5Cright)%5Cright)" alt="\left(u\circ s\right)\left(x\right)=u\left(s\left(x\right)\right)"/></div> <div> </div> <div>Esim. Muodosta</div> <div>a)<img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%5E2" alt="u\left(x\right)^2"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%5E2%3D3x%2B1" alt="s\left(x\right)^2=3x+1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%3Du%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%5Cleft(3x%2B1%5Cright)%5E2" alt="\left(u\circ s\right)\left(x\right)=u\left(s\left(x\right)\right)=u\left(3x+1\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)%3Ds%5Cleft(u%5Cleft(x%5Cright)%5Cright)%3Ds%5Cleft(x%5Cright)%5E2%3D3x%5E2%2B1" alt="\left(s\circ u\right)\left(x\right)=s\left(u\left(x\right)\right)=s\left(x\right)^2=3x^2+1"/><br/> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="u\left(x\right)=\cos x"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3Dx%5E2%2B1" alt="s\left(x\right)=x^2+1"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%3Du%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%5Cleft(x%5E2%2B1%5Cright)%3D%5Ccos%5Cleft(x%5E2%2B1%5Cright)" alt="\left(u\circ s\right)=u\left(s\left(x\right)\right)=u\left(x^2+1\right)=\cos\left(x^2+1\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cleft(s%5Ccirc%20u%5Cright)%3Ds%5Cleft(u%5Cleft(x%5Cright)%5Cright)%3Ds%5Cleft(%5Ccos%20x%5Cright)%3D%5Cleft(%5Ccos%20x%5Cright)%5E2%2B1%3D%5Ccos2x%2B1%5Cright)" alt="\left(\left(s\circ u\right)=s\left(u\left(x\right)\right)=s\left(\cos x\right)=\left(\cos x\right)^2+1=\cos2x+1\right)"/> <div><br/> <div> <div>Huom! yleensä </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%5Cne%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)" alt="\left(u\circ s\right)\left(x\right)\ne\left(s\circ u\right)\left(x\right)"/></div> </div> </div> <div>Esim.Tulkitse yhdistetyksi funktioksi </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft(3x%5E2%2B2x%5Cright)%5E2" alt="f\left(x\right)=\left(3x^2+2x\right)^2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D3x%5E2%2B2x" alt="s\left(x\right)=3x^2+2x"/><span>,</span><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E2" alt="u\left(x\right)=x^2"/><br/> <div>Tällöin</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%5Cleft(3x%5E2%2B2x%5Cright)%3D%5Cleft(3x%5E2%2B2x%5Cright)%5E2" alt="u\left(s\left(x\right)\right)=u\left(3x^2+2x\right)=\left(3x^2+2x\right)^2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7B1%2B%5Csin%20x%7D" alt="g\left(x\right)=\frac{3}{1+\sin x}"/></div> <div>Tapa 1:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D1%2B%5Csin%20x" alt="s\left(x\right)=1+\sin x"/>,<img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7Bx%7D" alt="u\left(x\right)=\frac{3}{x}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%5Cleft(%5Cfrac%7B3%7D%7B1%2B%5Csin%20x%7D%5Cright)" alt="u\left(s\left(x\right)\right)=u\left(\frac{3}{1+\sin x}\right)"/> <div>Tapa 2:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Csin%20x" alt="s\left(x\right)=\sin x"/>,<img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7B1%2Bx%7D" alt="u\left(x\right)=\frac{3}{1+x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3Du%5Cleft(%5Cfrac%7B3%7D%7B1%2B%5Csin%20x%7D%5Cright)" alt="u\left(s\left(x\right)\right)=u\left(\frac{3}{1+\sin x}\right)"/></div> 2018-12-04T13:50:25+02:00 https://peda.net/id/e391da24f2f 2018-11-29T12:47:15+02:00 <span>Lause</span> <div>Tangenttifunktiolle <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Ctan%20x" alt="f\left(x\right)=\tan x"/>pätee:</div> <div>- Funktio on määritelty, kun <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Cne%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x\ne\frac{\pi}{2}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div>- Arvojoukko on ]-∞,∞[ eli ℝ</div> <div>- Jatkuva määrittelyjoukossaan</div> <div>- Funktio on jaksollinen, perusjakso on π</div> <div>- Funktio on kasvava kaikilla määtittelyjoukkonsa osaväleillä</div> <div> </div> <div> Esim. Ratkaise yhtälö geogebralla arvioiden ja ilman apuvälineitä</div> <div> </div> <div>a) <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D2%2B%5Csqrt%5B%5D%7B3%7D" alt="\tan x=2+\sqrt[]{3}"/></div> <div>Geogebra: <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox1%7B%2C%7D31%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x\approx1{,}31+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div>MAOL: Eräs ratkaisu on <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B12%7D" alt="x=\frac{5\pi}{12}"/>. Kaikki ratkaisut ovat<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B12%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{5\pi}{12}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div>b) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan3x%3D%5Ctan%5Cfrac%7B4%5Cpi%7D%7B7%7D" alt="\tan3x=\tan\frac{4\pi}{7}"/> <div>Geogebralla: <img src="https://math-demo.abitti.fi/math.svg?latex=x%5Capprox0%7B%2C%7D6%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x\approx0{,}6+n\cdot\pi{,}\ n\in\mathbb{Z}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan3x%3D%5Ctan%5Cfrac%7B4%5Cpi%7D%7B7%7D" alt="\tan3x=\tan\frac{4\pi}{7}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cfrac%7B4%5Cpi%7D%7B7%7D" alt="3x=\frac{4\pi}{7}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B4%5Cpi%7D%7B21%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{4\pi}{21}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div><br/> Lause</div> <div>Jos<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Calpha" alt="x=\alpha"/>on yhtälön <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Calpha" alt="\tan x=\alpha"/> eräs ratkaisu, niin kaikki ratkaisut ovat <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Calpha%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\alpha+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div> </div> <div>Esim. Ratkaise yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Ccdot%5Csin%20x%3D-2%5Ccdot%5Ccos%5C%20x%5C%20%5Cleft%7C%5Cright%7C%3A%5Ccos%20x%7B%2C%7D%5C%20x%5Cne%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="4\cdot\sin x=-2\cdot\cos\ x\ \left|\right|:\cos x{,}\ x\ne\frac{\pi}{2}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D-2" alt="4\frac{\sin x}{\cos x}=-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\tan x=-\frac{1}{2}"/> <div> <div> <div>Jos</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi" alt="x=\frac{\pi}{2}+n\cdot\pi"/>, niin <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cpm1" alt="\sin x=\pm1"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D0" alt="\cos x=0"/> eli yhtälö on epätosi.</div> <div> </div> <div>Lause</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Ctan%20x%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D%3D1%2B%5Ctan%5E2x%7B%2C%7D%5C%20kun%5C%20x%5Cne%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7B%5Cmathbb%7BZ%7D%7D" alt="D\tan x=\frac{1}{\cos^2x}=1+\tan^2x{,}\ kun\ x\ne\frac{\pi}{2}+n\cdot\pi{,}\ n\in\mathbb{\mathbb{Z}}"/></div> Todistus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5C%20%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bg%5Cleft(x%5Cright)%7D%3D%5Cfrac%7Bf%27%5Cleft(x%5Cright)g%5Cleft(x%5Cright)-f%5Cleft(x%5Cright)g%27%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E%7B%5E2%7D%7D" alt="D\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{\left(g\left(x\right)\right)^{^2}}"/><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Csin%20x%3D%5Ccos%20x" alt="D\sin x=\cos x"/><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Ccos%20x%3D-%5Csin%20x" alt="D\cos x=-\sin x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Ctan%20x%3DD%5C%20%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B%5Ccos%20x%5Ccdot%5Ccos%20x%2B%5Csin%20x%5Ccdot%5Csin%20x%7D%7B%5Cleft(%5Ccos%20x%5Cright)%5E2%7D%3D%5Cfrac%7B%5Ccos%5E2x%2B%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D" alt="D\tan x=D\ \frac{\sin x}{\cos x}=\frac{\cos x\cdot\cos x+\sin x\cdot\sin x}{\left(\cos x\right)^2}=\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{1}{\cos^2x}"/><br/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Ccos%5E2x%2B%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%3D%5Cfrac%7B%5Ccos%5E2x%7D%7B%5Ccos%5E2x%7D%2B%5Cfrac%7B%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%3D1%2B%5Cfrac%7B%5Cleft(%5Csin%20x%5Cright)%5E2%7D%7B%5Cleft(%5Ccos%20x%5Cright)%5E%7B%5E2%7D%7D%3D1%2B%5Cleft(%5Ctan%20x%5Cright)%5E2%3D1%2B%5Ctan%5E2x" alt="\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{\cos^2x}{\cos^2x}+\frac{\sin^2x}{\cos^2x}=1+\frac{\left(\sin x\right)^2}{\left(\cos x\right)^{^2}}=1+\left(\tan x\right)^2=1+\tan^2x"/><br/> <br/> </div> </div> </div> 2018-11-28T12:39:31+02:00 https://peda.net/id/0efe534ef2f 2018-11-28T12:47:53+02:00 <div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Calpha%3Dy" alt="\sin\alpha=y"/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Calpha%3Dx" alt="\cos\alpha=x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Calpha%3D%5Cfrac%7By%7D%7Bx%7D" alt="\tan\alpha=\frac{y}{x}"/></div> </div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Calpha%3D%5Cfrac%7B%5Csin%5Calpha%7D%7B%5Ccos%5Calpha%7D" alt="\tan\alpha=\frac{\sin\alpha}{\cos\alpha}"/>, kun <img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha%5Cne90%C2%B0%2Bn%5Ccdot180%C2%B0" alt="\alpha\ne90°+n\cdot180°"/><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Calpha%5Cne%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D%5Cright)" alt="\left(\alpha\ne\frac{\pi}{2}+n\cdot\pi{,}\ n\in\mathbb{Z}\right)"/></div> 2018-11-28T12:47:53+02:00 https://peda.net/id/28ee90c0ed6 2018-11-21T10:48:04+02:00 <div>2.2 sinin ja kosinin derivaatat</div> <div> </div> <div>Lause </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Csin%20x%3D%5Ccos%20x" alt="D\sin x=\cos x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=D%5Ccos%20x%3D-%5Csin%20x" alt="D\cos x=-\sin x"/> <div><br/> <div>Esim. Määritä</div> <div>a) f'(x), kun f(x)=2sinx+cosx</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3DD2%5Csin%20x%2BD%5Ccos%20x%3D2D%5Csin%20x%2BD%5Csin%20x%3D2%5Ccdot%5Ccos%20x-%5Csin%20x" alt="f'\left(x\right)=D2\sin x+D\cos x=2D\sin x+D\sin x=2\cdot\cos x-\sin x"/></div> <div> </div> <div>b) f'(π/2), kun f(x)=sinx-3cosx</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(x%5Cright)%3DD%5Csin%20x-3D%5Ccos%20x%3D%5Ccos%20x%2B3%5Csin%20x" alt="f'\left(x\right)=D\sin x-3D\cos x=\cos x+3\sin x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%27%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D%5Ccos%5Cfrac%7B%5Cpi%7D%7B2%7D%2B3%5Csin%5Cfrac%7B%5Cpi%7D%7B2%7D%3D0%2B3%3D3" alt="f'\left(\frac{\pi}{2}\right)=\cos\frac{\pi}{2}+3\sin\frac{\pi}{2}=0+3=3"/><br/> <br/> Esim. Derivoi</div> <div>a) h(x)=x³cosx</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cleft(f%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%5Cright)%3Df%27%5Cleft(x%5Cright)g%5Cleft(x%5Cright)%2Bg%27%5Cleft(x%5Cright)%5Ccdot%20f%5Cleft(x%5Cright)" alt="D\left(f\left(x\right)g\left(x\right)\right)=f'\left(x\right)g\left(x\right)+g'\left(x\right)\cdot f\left(x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=h%27%5Cleft(x%5Cright)%3D3x%5E2%5Ccos%20x-x%5E3%5Csin%20x" alt="h'\left(x\right)=3x^2\cos x-x^3\sin x"/></div> <div> </div> <div>b) <img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D" alt="h\left(x\right)=\frac{\sin x}{\cos x}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=D%5Cfrac%7Bf%5Cleft(x%5Cright)%7D%7Bg%5Cleft(x%5Cright)%7D%3D%5Cfrac%7Bf%27%5Cleft(x%5Cright)g%5Cleft(x%5Cright)-g%27%5Cleft(x%5Cright)f%5Cleft(x%5Cright)%7D%7B%5Cleft(g%5Cleft(x%5Cright)%5Cright)%5E2%7D%5C%20%7B%2C%7Dg%5Cleft(x%5Cright)%5Cne0" alt="D\frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)g\left(x\right)-g'\left(x\right)f\left(x\right)}{\left(g\left(x\right)\right)^2}\ {,}g\left(x\right)\ne0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%27%5Cleft(x%5Cright)%3D%5Cfrac%7B%5Ccos%20x%5Ccdot%5Ccos%20x%2B%5Csin%20x%5Ccdot%5Csin%20x%7D%7B%5Cleft(%5Ccos%20x%5Cright)%5E2%7D%3D%5Cfrac%7B%5Ccos%5E2x%2B%5Csin%5E2x%7D%7B%5Ccos%5E2x%7D%3D%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D" alt="h'\left(x\right)=\frac{\cos x\cdot\cos x+\sin x\cdot\sin x}{\left(\cos x\right)^2}=\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{1}{\cos^2x}"/><br/> <br/> <div> </div> <div> </div> </div> 2018-11-21T10:48:04+02:00 https://peda.net/id/b32b7b14ed1 2018-11-21T01:47:54+02:00 <div>Molemmille funktiolle Esim. Määritä funktion arvojoukko, kun </div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%20x" alt="f\left(x\right)=\sin x"/><span>ja </span><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="g\left(x\right)=\cos x"/><span>pätee:</span> <div>- Funktio on määritelty x∈ℝ</div> <div>- Funktion arvojoukko on [-1,1]</div> <div>- Funktio on jatkuva</div> <div>- Funktion kuvaaja toistuu samanlaisena 2π:n välein</div> <div> </div> <div>Esim Määritä funktion joukko, kun<br/> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3%5Csin%20x-2" alt="f\left(x\right)=3\sin x-2"/></div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Csin%20x%5Cle1%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot3" alt="-1\le\sin x\le1\ \ \ \ \ \left|\right|\cdot3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle3%5Csin%20x%5Cle3%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C-2" alt="-3\le3\sin x\le3\ \ \ \ \ \left|\right|-2"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-5%5Cle3%5Csin%20x-2%5Cle1" alt="-5\le3\sin x-2\le1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-5%5Cle%20f%5Cleft(x%5Cright)%5Cle1" alt="-5\le f\left(x\right)\le1"/> <div>eli arvojoukko on [-5,1]</div> <div> </div> <div>Lause </div> <div>Funktioiden </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Csin%5Cleft(Cx%2BD%5Cright)" alt="f\left(x\right)=\sin\left(Cx+D\right)"/>ja <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Ccos%5Cleft(Cx%2BD%5Cright)" alt="g\left(x\right)=\cos\left(Cx+D\right)"/>, missä C&gt;1, perusjakso on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%5Cpi%7D%7BC%7D" alt="\frac{2\pi}{C}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D%5Ccos%20x" alt="\sin\left(x+\frac{\pi}{2}\right)=\cos x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D%5Csin%20x" alt="\cos\left(x+\frac{\pi}{2}\right)=\sin x"/></div> <div> </div> <div>214</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D3%5Csin2x%2B5" alt="f\left(x\right)=3\sin2x+5"/></div> <div>Arvo kohdassa x=0</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D3%5Ccdot%5Csin%5Cleft(2%5Ccdot0%5Cright)%2B5%3D3%5Ccdot0%2B5%3D5" alt="f\left(0\right)=3\cdot\sin\left(2\cdot0\right)+5=3\cdot0+5=5"/></div> <span>Nollakohdat: Ratkaistaan yhtälö </span><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D0" alt="f\left(x\right)=0"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ccdot%5Csin2x%2B5%3D0" alt="3\cdot\sin2x+5=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3%5Ccdot%5Csin2x%3D-5" alt="3\cdot\sin2x=-5"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2x%3D-%5Cfrac%7B5%7D%7B3%7D" alt="\sin2x=-\frac{5}{3}"/><br/> <div>sinin arvot ovat välillä [-1,1], joten yhtälöllä ei ole ratkaisua, eikä funktiolla ole nollakohtia.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Csin%20x%5Cle1" alt="-1\le\sin x\le1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-1%5Cle%5Csin2x%5Cle1%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot3" alt="-1\le\sin2x\le1\ \ \ \ \ \left|\right|\cdot3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-3%5Cle3%5Csin2x%5Cle3%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%2B5" alt="-3\le3\sin2x\le3\ \ \ \ \ \left|\right|+5"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cle3%5Csin2x%2B5%5Cle8" alt="2\le3\sin2x+5\le8"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cle%20f%5Cleft(x%5Cright)%5Cle8" alt="2\le f\left(x\right)\le8"/></div> <div>Arvojoukko on [2,8]</div> 2018-11-21T01:47:54+02:00 https://peda.net/id/b7fd19aee7e 2018-11-15T11:09:45+02:00 <div>Esim. Ratkaise kaikki kulmat x, jolle <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D" alt="\sin x=\frac{1}{\sqrt[]{2}}"/></div> <div>MAOL: Kulmat <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="x=\frac{\pi}{4}"/>on yhtälön eräs ratkaisu</div> <div>koska <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%5Csin%5Cleft(%5Cpi-%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)" alt="\sin\frac{\pi}{4}=\sin\left(\pi-\frac{\pi}{4}\right)"/>, niin myös <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D" alt="x=\pi-\frac{\pi}{4}=\frac{3\pi}{4}"/>on yhtälön ratkaisu</div> <div>kaikki ratkaisut ovat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5C%20%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{\pi}{4}+n\cdot2\ \pi{,}\ n\in\mathbb{Z}"/> tai <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{3\pi}{4}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/></div> <div>Lause: Jos yhtälöllä<img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Calpha" alt="\sin x=\alpha"/> on yksi ratkaisu x=α, niin kaikki ratkaisut ovat <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Calpha%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7B%5Cmathbb%7B%5Cmathbb%7B%5Cmathbb%7B%5Cmathbb%7BZ%7D%7D%7D%7D%7D" alt="x=\alpha+n\cdot2\pi{,}\ n\in\mathbb{\mathbb{\mathbb{\mathbb{\mathbb{Z}}}}}"/>tai <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-%5Calpha%2Bn%5Ccdot2%5Cpi" alt="x=\pi-\alpha+n\cdot2\pi"/></div> <div> </div> <div>Esim. Ratkaise yhtälö <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B2%7D%5Ccos%20x%2B1%3D0" alt="\sqrt[]{2}\cos x+1=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B2%7D%5Ccos%20x%2B1%3D0" alt="\sqrt[]{2}\cos x+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csqrt%5B%5D%7B2%7D%5Ccos%20x%3D-1" alt="\sqrt[]{2}\cos x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D" alt="\cos x=-\frac{1}{\sqrt[]{2}}"/><br/> <div>MAOL: yksi ratkaisu on <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D" alt="x=\frac{3\pi}{4}"/></div> <div>koska </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D%5Ccos%5Cleft(-%5C%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%5Cright)" alt="\cos\frac{3\pi}{4}=\cos\left(-\ \frac{3\pi}{4}\right)"/>, niin <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D" alt="x=-\frac{3\pi}{4}"/>on ratkaisu<br/> <div>Kaikki ratkaisut ovat <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{3\pi}{4}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=-\frac{3\pi}{4}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/></div> <div> </div> <div>Lause: Jos yhtälöllä </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Calpha" alt="\cos x=\alpha"/>on yksi ratkaisu x=α, niin kaikki ratkaisut ovat <span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Calpha%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\alpha+n\cdot2\pi{,}\ n\in\mathbb{Z}"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Calpha%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=-\alpha+n\cdot2\pi{,}\ n\in\mathbb{Z}"/><br/>  <br/> <div>t.173</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%5Cfrac%7Bx%7D%7B3%7D%2B%5Csqrt%5B%5D%7B2%7D%3D0" alt="2\sin\frac{x}{3}+\sqrt[]{2}=0"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D-%5Csqrt%5B%5D%7B2%7D" alt="2\sin\frac{x}{3}=-\sqrt[]{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B%5Csqrt%5B%5D%7B2%7D%7D%7B2%7D" alt="\sin\frac{x}{3}=-\frac{\sqrt[]{2}}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B2%7D%7B2%5Csqrt%5B%5D%7B2%7D%7D" alt="\sin\frac{x}{3}=-\frac{2}{2\sqrt[]{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D" alt="\sin\frac{x}{3}=-\frac{1}{\sqrt[]{2}}"/>'<br/> <div>MAOL: Kulman<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%5Cpi%7D%7B4%7D" alt="\frac{5\pi}{4}"/> sini on <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D" alt="-\frac{1}{\sqrt[]{2}}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%5B%5D%7B2%7D%7D" alt="\sin\frac{x}{3}=-\frac{1}{\sqrt[]{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cfrac%7B5%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi" alt="\frac{x}{3}=\frac{5\pi}{4}+n\cdot2\pi"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B3%7D%3D%5Cpi-%5Cfrac%7B5%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="\frac{x}{3}=\pi-\frac{5\pi}{4}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B15%5Cpi%7D%7B4%7D%2Bn%5Ccdot6%5Cpi" alt="x=\frac{15\pi}{4}+n\cdot6\pi"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot6%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=-\frac{3\pi}{4}+n\cdot6\pi{,}\ n\in\mathbb{Z}"/></div> <br/> <div>Selvitetään, mitkä ratkaisut ovat välillä ]-12π,12π[ eli ]-48π,48π[</div> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D0%3A%5C%20x%3D%5Cfrac%7B15%5Cpi%7D%7B4%7D%5C%20tai%5C%20x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D" alt="n=0:\ x=\frac{15\pi}{4}\ tai\ x=-\frac{3\pi}{4}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n%3D1%3A%5C%20x%3D%5Cfrac%7B15%5Cpi%7D%7B4%7D%2B6%5Cpi%3D%5Cfrac%7B39%5Cpi%7D%7B4%7D%5C%20tai%5C%20x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2B6%5Cpi%3D%5Cfrac%7B21%5Cpi%7D%7B4%7D" alt="n=1:\ x=\frac{15\pi}{4}+6\pi=\frac{39\pi}{4}\ tai\ x=-\frac{3\pi}{4}+6\pi=\frac{21\pi}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D-1%3A%5C%20x%3D-%5Cfrac%7B9%5Cpi%7D%7B4%7D%5C%20tai%5C%20x%3D-%5Cfrac%7B27%5Cpi%7D%7B4%7D" alt="n=-1:\ x=-\frac{9\pi}{4}\ tai\ x=-\frac{27\pi}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D2%3A%5C%20x%3D%5Cfrac%7B63%5Cpi%7D%7B4%7D%5Cleft(hyl.%5Cright)%5C%20tai%5C%20x%3D%5Cfrac%7B45%5Cpi%7D%7B4%7D" alt="n=2:\ x=\frac{63\pi}{4}\left(hyl.\right)\ tai\ x=\frac{45\pi}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D-2%3A%5C%20x%3D%5Cfrac%7B33%5Cpi%7D%7B4%7D%5C%20tai%5C%20x%3D-%5Cfrac%7B59%5Cpi%7D%7B4%7D%5Cleft(hyl.%5Cright)" alt="n=-2:\ x=\frac{33\pi}{4}\ tai\ x=-\frac{59\pi}{4}\left(hyl.\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D-3%3A%5C%20x%3D-%5Cfrac%7B57%5Cpi%7D%7B4%7D%5Cleft(hyl.%5Cright)" alt="n=-3:\ x=-\frac{57\pi}{4}\left(hyl.\right)"/></div> <div>Ratkaisutista halutulle välille kuuluvat</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B33%5Cpi%7D%7B4%7D%7B%2C%7D-%5Cfrac%7B27%5Cpi%7D%7B4%7D%7B%2C%7D-%5Cfrac%7B9%5Cpi%7D%7B4%7D%7B%2C%7D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B15%5Cpi%7D%7B4%7D%7B%2C%7D%5Cfrac%7B21%5Cpi%7D%7B4%7D%7B%2C%7D%5Cfrac%7B39%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B45%5Cpi%7D%7B4%7D" alt="-\frac{33\pi}{4}{,}-\frac{27\pi}{4}{,}-\frac{9\pi}{4}{,}-\frac{3\pi}{4}{,}\ \frac{15\pi}{4}{,}\frac{21\pi}{4}{,}\frac{39\pi}{4}{,}\ \frac{45\pi}{4}"/><br/> <br/> <div>Esim. Ratkaise </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D-%5Csin2x" alt="\sin\left(x+\frac{\pi}{2}\right)=-\sin2x"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)%3D%5Csin%5Cleft(-2x%5Cright)" alt="\sin\left(x+\frac{\pi}{2}\right)=\sin\left(-2x\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-2x%2Bn%5Ccdot2%5Cpi" alt="x+\frac{\pi}{2}=-2x+n\cdot2\pi"/><span>tai </span><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cpi%2B2x%2Bn%5Ccdot2%5Cpi" alt="x+\frac{\pi}{2}=\pi+2x+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-2x%2Bn%5Ccdot2%5Cpi" alt="x+\frac{\pi}{2}=-2x+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%2B2x%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x+2x=-\frac{\pi}{2}+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="3x=-\frac{\pi}{2}+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot%5Cfrac%7B2%7D%7B3%7D%5Cpi" alt="x=-\frac{\pi}{6}+n\cdot\frac{2}{3}\pi"/> <div><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cpi%2B2x%2Bn%5Ccdot2%5Cpi" alt="x+\frac{\pi}{2}=\pi+2x+n\cdot2\pi"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x-2x%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x-2x=\pi-\frac{\pi}{2}+n\cdot2\pi"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="-x=\frac{\pi}{2}+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi" alt="x=-\frac{\pi}{2}+n\cdot2\pi"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot%5Cfrac%7B2%7D%7B3%7D%5Cpi" alt="x=-\frac{\pi}{6}+n\cdot\frac{2}{3}\pi"/> tai <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D%2Bn%5Ccdot2%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=-\frac{\pi}{2}+n\cdot2\pi{,}\ n\in\mathbb{Z}"/></div> <br/> <div> </div> </div> 2018-11-14T11:07:31+02:00 https://peda.net/id/6023ebbae73 2018-11-13T13:22:16+02:00 <div>Esim. Laske ilmna laskinta </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cleft(-240%C2%B0%5Cright)%3D-%5Csin240%C2%B0%3D-%5Cleft(-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D%5Cright)%3D%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\sin\ \left(-240°\right)=-\sin240°=-\left(-\frac{\sqrt[]{3}}{2}\right)=\frac{\sqrt[]{3}}{2}"/></div> <span>b)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(-%5Cfrac%7B7%5Cpi%7D%7B6%7D%5Cright)%3D%5Ccos%5Cleft(%5Cfrac%7B7%5Cpi%7D%7B6%7D%5Cright)%3D-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\cos\left(-\frac{7\pi}{6}\right)=\cos\left(\frac{7\pi}{6}\right)=-\frac{\sqrt[]{3}}{2}"/><br/> <div>c) Osoita, että <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7B%5Cpi%7D%7B5%7D%3D%5Csin%5Cfrac%7B4%5Cpi%7D%7B5%7D" alt="\sin\frac{\pi}{5}=\sin\frac{4\pi}{5}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cfrac%7B%5Cpi%7D%7B5%7D%3D%5Csin%5Cleft(%5Cpi-%5Cfrac%7B%5Cpi%7D%7B5%7D%5Cright)%3D%5Csin%5Cleft(%5Cfrac%7B5%5Cpi%7D%7B5%7D-%5Cfrac%7B%5Cpi%7D%7B5%7D%5Cright)%3D%5Csin%5Cfrac%7B4%5Cpi%7D%7B5%7D" alt="\sin\frac{\pi}{5}=\sin\left(\pi-\frac{\pi}{5}\right)=\sin\left(\frac{5\pi}{5}-\frac{\pi}{5}\right)=\sin\frac{4\pi}{5}"/></div> <br/> <div>Lause: Kaksinkertaisen kulman sini ja kosini</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2%5Calpha%3D2%5Csin%5Calpha%5Ccos%5Calpha" alt="\sin2\alpha=2\sin\alpha\cos\alpha"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2%5Calpha%3D%5Ccos%5E2%5Calpha-%5Csin%5E2%5Calpha%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft(%5Csin%5E2%5Calpha%2B%5Ccos%5E2%5Calpha%3D1%5Cright)" alt="\cos2\alpha=\cos^2\alpha-\sin^2\alpha\ \ \ \ \ \left(\sin^2\alpha+\cos^2\alpha=1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2%5Calpha%3D1-2%5Csin%5E2%5Calpha" alt="\cos2\alpha=1-2\sin^2\alpha"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2%5Calpha%3D2%5Ccos%5E2%5Calpha-1" alt="\cos2\alpha=2\cos^2\alpha-1"/><br/> <div> </div> <div>Esim. Tidetään, että<img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Calpha%3D%5Cfrac%7B4%7D%7B5%7D" alt="\cos\alpha=\frac{4}{5}"/>. Määritä cos2α ja sin2α</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2%5Calpha%3D2%5Ccdot%5Cleft(%5Cfrac%7B4%7D%7B5%7D%5Cright)%5E2-1%3D2%5Ccdot%5Cfrac%7B16%7D%7B25%7D%3D%5Cfrac%7B32%7D%7B25%7D-1%3D1%5Cfrac%7B7%7D%7B25%7D-1%3D%5Cfrac%7B7%7D%7B25%7D%3D0%7B%2C%7D28" alt="\cos2\alpha=2\cdot\left(\frac{4}{5}\right)^2-1=2\cdot\frac{16}{25}=\frac{32}{25}-1=1\frac{7}{25}-1=\frac{7}{25}=0{,}28"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csin%5Calpha%5Cright)%5E2%2B%5Cleft(%5Ccos%5Calpha%5Cright)%5E2%3D1" alt="\left(\sin\alpha\right)^2+\left(\cos\alpha\right)^2=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csin%5Calpha%5Cright)%5E2%3D1-%5Cleft(%5Ccos%5Calpha%5Cright)%5E2" alt="\left(\sin\alpha\right)^2=1-\left(\cos\alpha\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csin%5Calpha%5Cright)%5E2%3D1-%5Cleft(%5Cfrac%7B4%7D%7B5%7D%5Cright)%5E2" alt="\left(\sin\alpha\right)^2=1-\left(\frac{4}{5}\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csin%5Calpha%5Cright)%5E2%3D1-%5Cfrac%7B16%7D%7B25%7D" alt="\left(\sin\alpha\right)^2=1-\frac{16}{25}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Csin%5Calpha%5Cright)%5E2%3D%5Cfrac%7B9%7D%7B25%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Csqrt%5B%5D%7B%7D" alt="\left(\sin\alpha\right)^2=\frac{9}{25}\ \ \ \ \ \left|\right|\sqrt[]{}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Calpha%3D%5Cpm%5Cfrac%7B3%7D%7B5%7D%3D%5Cpm0%7B%2C%7D6" alt="\sin\alpha=\pm\frac{3}{5}=\pm0{,}6"/><br/> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2%5Calpha%3D2%5Csin%5Calpha%5Ccos%5Calpha" alt="\sin2\alpha=2\sin\alpha\cos\alpha"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2%5Calpha%3D2%5Ccdot%5Cfrac%7B3%7D%7B5%7D%5Ccdot%5Cfrac%7B4%7D%7B5%7D%3D0%7B%2C%7D96" alt="\sin2\alpha=2\cdot\frac{3}{5}\cdot\frac{4}{5}=0{,}96"/><br/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2%5Calpha%3D2%5Ccdot%5Cleft(-%5Cfrac%7B3%7D%7B5%7D%5Cright)%5Ccdot%5Cfrac%7B4%7D%7B5%7D%3D-0%7B%2C%7D96" alt="\sin2\alpha=2\cdot\left(-\frac{3}{5}\right)\cdot\frac{4}{5}=-0{,}96"/></div> 2018-11-13T13:22:16+02:00 https://peda.net/id/ecafc250e33 2018-11-08T11:23:16+02:00 <span>Määritä taulukkokirjan avulla</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin30%C2%B0" alt="\sin30°"/></div> <div> </div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)" alt="\cos\left(-\frac{\pi}{6}\right)"/></div> <div>Etsitään välin [0,2π] kulma, jolla on sama kehäpiste kuin kulmalla<img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}"/>. Lisätään kulmaan<img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}"/>täysiä kulmia, kunnes saadaan postiivinen kulma.</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%2B2%5Cpi%3D%5Cfrac%7B11%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}+2\pi=\frac{11\pi}{6}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D%5Ccos%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%2B2%5Cpi%5Cright)%3D%5Ccos%5Cfrac%7B11%5Cpi%7D%7B6%7D%3D%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\cos\left(-\frac{\pi}{6}\right)=\cos\left(-\frac{\pi}{6}+2\pi\right)=\cos\frac{11\pi}{6}=\frac{\sqrt[]{3}}{2}"/><br/> <div> <div> </div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cfrac%7B17%5Cpi%7D%7B6%7D%3D%5Ccos%5Cfrac%7B17%7D%7B6%7D%5Cpi%3D%5Ccos2%5Cfrac%7B5%7D%7B6%7D%5Cpi%3D%5Ccos%5Cleft(2%5Cpi%2B%5Cfrac%7B5%7D%7B6%7D%5Cpi%5Cright)%3D%5Ccos%5Cfrac%7B5%5Cpi%7D%7B6%7D%3D-%5Cfrac%7B%5Csqrt%5B%5D%7B3%7D%7D%7B2%7D" alt="\cos\frac{17\pi}{6}=\cos\frac{17}{6}\pi=\cos2\frac{5}{6}\pi=\cos\left(2\pi+\frac{5}{6}\pi\right)=\cos\frac{5\pi}{6}=-\frac{\sqrt[]{3}}{2}"/></div> </div> 2018-11-08T11:23:16+02:00 https://peda.net/id/28631e06e26 2018-11-07T12:42:17+02:00 <span>Määritelmä: </span><br/> <span>Kulman suurus radiaaneina on kulmaa vastaavan ympyrän kaaren pituus kun ympyrän säde on 1 ja ympyrän keskipiste on kulma kärki.</span><br/> <div> <br/> <a href="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/esimerkki-1-png#top" title="Esimerkki 1.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/esimerkki-1-png:file/photo/7d14653c9204ebda7695887728ec5faaa085f75f/Esimerkki%201.PNG" alt="" title="Esimerkki 1.PNG" class="inline" loading="lazy"/></a><br/> <br/> </div> <div>Esim. Munna</div> <div>a)<br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cpi%7D%7B6%7D%5C%20" alt="\frac{\pi}{6}\ "/> asteiksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cpi%7D%7B6%7D%3D%5Cfrac%7B180%C2%B0%7D%7B6%7D%3D30%C2%B0" alt="\frac{\pi}{6}=\frac{180°}{6}=30°"/></div> <div> </div> <div>b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="\frac{5\pi}{6}"/> asteiksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%5Cpi%7D%7B6%7D%3D5%5Ccdot%5Cfrac%7B%5Cpi%7D%7B6%7D%3D5%5Ccdot30%C2%B0%3D150%C2%B0" alt="\frac{5\pi}{6}=5\cdot\frac{\pi}{6}=5\cdot30°=150°"/></div> <div> </div> <div>c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%7B%2C%7D4%5C%20%5Cleft(rad%5Cright)" alt="2{,}4\ \left(rad\right)"/>asteiksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%3D180%C2%B0" alt="\pi=180°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%3D%5Cfrac%7B180%C2%B0%7D%7B%5Cpi%7D" alt="1=\frac{180°}{\pi}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%7B%2C%7D4%3D%5Cfrac%7B180%C2%B0%7D%7B%5Cpi%7D%5Ccdot2%7B%2C%7D4" alt="2{,}4=\frac{180°}{\pi}\cdot2{,}4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%7B%2C%7D4%5Capprox138%C2%B0" alt="2{,}4\approx138°"/></div> <div><br/> <div>d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=240%C2%B0" alt="240°"/>radiaaneiksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=180%C2%B0%3D%5Cpi" alt="180°=\pi"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%C2%B0%3D%5Cfrac%7B%5Cpi%7D%7B180%C2%B0%7D" alt="1°=\frac{\pi}{180°}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=240%C2%B0%3D%5Cfrac%7B%5Cpi%7D%7B180%C2%B0%7D%5Ccdot240%C2%B0" alt="240°=\frac{\pi}{180°}\cdot240°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=240%C2%B0%3D%5Cfrac%7B4%5Cpi%7D%7B3%7D" alt="240°=\frac{4\pi}{3}"/><br/> <br/> - Suunnatulla kulmalla on alku- ja loppukylki</div> <ul> <li>Positiivisen kulman kiertosuunta on vastapäivään</li> <li>Negatiivisen kulma kiertosuunta on myötäpäivään</li> </ul> - Kun kulman piirrettään yksikköympyrälle (säde on 1, kp (0,0)) <ul> <li>Kulman kärki on origossa</li> <li>Alkukylki on posit. x-akseli</li> <li>Loppukyljen- ja ympyrän lp. on kehäpiste</li> </ul> <a href="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/esimerkki-2-png#top" title="Esimerkki 2.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/esimerkki-2-png:file/photo/6e3ce7f7d2ea8756382cd13575289e10c0371337/Esimerkki%202.PNG" alt="" title="Esimerkki 2.PNG" class="inline" loading="lazy"/></a><br/> <div> </div> <div>Esim. Piirrä yksikköympyrälle sueraavat kulmat ja merkitse vastaavat kehäpisteet</div> <div>a) </div> <img src="https://math-demo.abitti.fi/math.svg?latex=405%C2%B0" alt="405°"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-a-png#top" title="Rad a.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-a-png:file/photo/16248dbc4133021cc0d831b8127dd7396956e35a/Rad%20a.PNG" alt="" title="Rad a.PNG" class="inline" loading="lazy"/></a><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-120%C2%B0" alt="-120°"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-b-png#top" title="Rad b.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-b-png:file/photo/3c521dc7267df7127532d43ed9397413d3303f03/Rad%20b.PNG" alt="" title="Rad b.PNG" class="inline" loading="lazy"/></a><br/> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%5Cpi%7D%7B4%7D%5Cleft(rad%5Cright)" alt="\frac{3\pi}{4}\left(rad\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%3D180%C2%B0" alt="\pi=180°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D3%5Ccdot%5Cfrac%7B%5Cpi%7D%7B4%7D%3D3%5Ccdot%5Cfrac%7B180%C2%B0%7D%7B4%7D%3D135%C2%B0" alt="\frac{3\pi}{4}=3\cdot\frac{\pi}{4}=3\cdot\frac{180°}{4}=135°"/><br/> <a href="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-c-png#top" title="Rad c.PNG"><img src="https://peda.net/p/kirin_porsti/ma/ma7p/esimerkit/kpl-1-1/rad-c-png:file/photo/1445361820c74822be81f467a2d60e65caa8dcf7/Rad%20c.PNG" alt="" title="Rad c.PNG" class="inline" loading="lazy"/></a> <div><br/> <div> </div> </div> </div> <br/> <br/> 2018-11-07T11:26:08+02:00