https://peda.net/id/6b9cd1180aa 2019-11-19T10:19:22+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/b66e5e020b7 2019-11-20T11:39:56+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%20x-1%3D0" alt="2\sin x-1=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin x=\frac{1}{2}"/></div> <div>yksi ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="x=\frac{\pi}{6}"/><!--filtered attribute: style="display: inline;"--></div> <div>täydellinen ratkaisu</div> <div><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/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot2%5Cpi" alt="x=\pi-\frac{\pi}{6}+n\cdot2\pi"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2x-%5Csin%5Cfrac%7B2%5Cpi%7D%7B7%7D%3D0" alt="\sin2x-\sin\frac{2\pi}{7}=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin2x%3D%5Csin%5Cfrac%7B2%5Cpi%7D%7B7%7D" alt="\sin2x=\sin\frac{2\pi}{7}"/></div> <div>ei löydy taulukosta, pitää laskea laskimella likiarvo</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D0%7B%2C%7D78183..." alt="2x=0{,}78183..."/><!--filtered attribute: style="display: inline;"--></div> <div>yksi ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%7B%2C%7D3909..." alt="x=0{,}3909..."/></div> <div>täydellinen ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%7B%2C%7D3909...%2Bn%5Ccdot2%5Cpi" alt="x=0{,}3909...+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cpi-0%7B%2C%7D3909...%2Bn%5Ccdot2%5Cpi" alt="x=\pi-0{,}3909...+n\cdot2\pi"/></div> <br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos4x%2B1%3D0" alt="2\cos4x+1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos4x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\cos4x=-\frac{1}{2}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="4x=\frac{2\pi}{3}"/></div> yksi ratkaisu<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="x=\frac{\pi}{6}"/></div> <div>täydellinen ratkaisu</div> <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/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><!--filtered attribute: style="display: inline;"--><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/> d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(1-x%5Cright)-1%3D0" alt="\cos\left(1-x\right)-1=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(1-x%5Cright)%3D1" alt="\cos\left(1-x\right)=1"/><br/> <div>yksi ratkaisu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1-x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="1-x=\frac{\pi}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D-1" alt="-x=\frac{\pi}{4}-1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1-%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="x=1-\frac{\pi}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D" alt="x=\frac{3\pi}{4}"/><br/> täydellinen ratkaisu<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi" alt="x=\frac{3\pi}{4}+n\cdot2\pi"/><br/> tai<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot2%5Cpi" alt="x=-\frac{3\pi}{4}+n\cdot2\pi"/></div> 2019-11-20T11:39:56+02:00 https://peda.net/id/d22872160b7 2019-11-20T11:19:14+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos4x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="\cos4x=-\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=4x%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="4x=\frac{2\pi}{3}+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%7B%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{6}+n\cdot\frac{\pi}{2}"/><br/> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=4x%3D-%5Cfrac%7B2%5Cpi%7D%7B3%7D%2Bn%5Ccdot2%5Cpi" alt="4x=-\frac{2\pi}{3}+n\cdot2\pi"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="x=-\frac{\pi}{6}+n\cdot\frac{\pi}{2}"/></div> <div>b) mitkä ratkaisuista ovat välillä ]-π,π[</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot%5Cleft(-6%5Cright)%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B3%5Cpi%7D%7B2%7D" alt="-\frac{\pi}{6}\cdot\left(-6\right)+\frac{\pi}{2}=\frac{3\pi}{2}"/> hylätään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot%5Cleft(-3%5Cright)%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cpi" alt="-\frac{\pi}{6}\cdot\left(-3\right)+\frac{\pi}{2}=\pi"/> hylätään</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot%5Cleft(-2%5Cright)%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}\cdot\left(-2\right)+\frac{\pi}{2}=\frac{5\pi}{6}"/> hyväksytään</div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot%5Cleft(-1%5Cright)%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="-\frac{\pi}{6}\cdot\left(-1\right)+\frac{\pi}{2}=\frac{2\pi}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot0%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="-\frac{\pi}{6}\cdot0+\frac{\pi}{2}=\frac{\pi}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot1%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="-\frac{\pi}{6}\cdot1+\frac{\pi}{2}=\frac{\pi}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot2%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}\cdot2+\frac{\pi}{2}=\frac{\pi}{6}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot3%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D0" alt="-\frac{\pi}{6}\cdot3+\frac{\pi}{2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot4%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cfrac%7B%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}\cdot4+\frac{\pi}{2}=-\frac{\pi}{6}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot5%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="-\frac{\pi}{6}\cdot5+\frac{\pi}{2}=-\frac{\pi}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot6%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="-\frac{\pi}{6}\cdot6+\frac{\pi}{2}=-\frac{\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot7%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cfrac%7B2%5Cpi%7D%7B3%7D" alt="-\frac{\pi}{6}\cdot7+\frac{\pi}{2}=-\frac{2\pi}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot8%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="-\frac{\pi}{6}\cdot8+\frac{\pi}{2}=-\frac{5\pi}{6}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Ccdot9%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%3D-%5Cpi" alt="-\frac{\pi}{6}\cdot9+\frac{\pi}{2}=-\pi"/> hylätään</div> <div> </div> ratkaisuista välillä ]-π,π[ ovat <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B5%5Cpi%7D%7B6%7D%7B%2C%7D%5C%20-%5Cfrac%7B2%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D-%5Cfrac%7B%5Cpi%7D%7B3%7D%7B%2C%7D-%5Cfrac%7B%5Cpi%7D%7B6%7D%7B%2C%7D%5C%200%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B6%7D%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B2%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="-\frac{5\pi}{6}{,}\ -\frac{2\pi}{3}{,}\ -\frac{\pi}{2}{,}-\frac{\pi}{3}{,}-\frac{\pi}{6}{,}\ 0{,}\ \frac{\pi}{6}{,}\ \frac{\pi}{3}{,}\ \frac{\pi}{2}{,}\ \frac{2\pi}{3}{,}\ \frac{5\pi}{6}"/></div> 2019-11-20T11:19:14+02:00 https://peda.net/id/a06301f80b6 2019-11-20T10:35:18+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin3x%3D%5Csin2x" alt="\sin3x=\sin2x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D2x%2Bn2%5Cpi%5C%20tai%5C%203x%3D%5Cpi-2x%2Bn2%5Cpi" alt="3x=2x+n2\pi\ tai\ 3x=\pi-2x+n2\pi"/></div> <br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Ccdot2%5Cpi%5C%20" alt="x=n\cdot2\pi\ "/></div> <div>tai</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=5x%3D%5Cpi%2Bn%5Ccdot2%5Cpi" alt="5x=\pi+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B5%7D%2Bn%5Ccdot%5Cfrac%7B2%5Cpi%7D%7B5%7D%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{\pi}{5}+n\cdot\frac{2\pi}{5}{,}\ n\in\mathbb{Z}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2x-%5Ccos4x%3D0" alt="\cos2x-\cos4x=0"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2x%3D%5Ccos4x" alt="\cos2x=\cos4x"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D4x%2Bn%5Ccdot2%5Cpi" alt="2x=4x+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2x%3Dn%5Ccdot2%5Cpi" alt="-2x=n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Cpi" alt="x=n\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=tai" alt="tai"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D-4x%2Bn%5Ccdot2%5Cpi" alt="2x=-4x+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6x%3Dn%5Ccdot2%5Cpi" alt="6x=n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=n\cdot\frac{\pi}{3}"/><br/> ratkaisu <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Cpi" alt="x=n\pi"/> tarkoittaa kulmia ..., -2π, -π, 0, π, 2π, ...<br/> ratkaisu <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=n\cdot\frac{\pi}{3}"/> tarkoittaa kulmia <img src="https://math-demo.abitti.fi/math.svg?latex=...%7B%2C%7D%5C%20-%5Cfrac%7B%5Cpi%7D%7B3%7D%7B%2C%7D%5C%200%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B2%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cpi%7B%2C%7D%5C%20..." alt="...{,}\ -\frac{\pi}{3}{,}\ 0{,}\ \frac{\pi}{3}{,}\ \frac{2\pi}{3}{,}\ \pi{,}\ ..."/> eli sisältää ratkaisun <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Cpi" alt="x=n\pi"/> kulmat<br/> yhtälön ratkaisu on siis <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=n\cdot\frac{\pi}{3}"/> 2019-11-20T10:27:44+02:00 https://peda.net/id/17d9ad8c0ab 2019-11-19T11:43:51+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-0%7B%2C%7D55" alt="\cos x=-0{,}55"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D3%7B%2C%7D72" alt="x=3{,}72"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20x%3D0%7B%2C%7D71" alt="\sin\ x=0{,}71"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D0%7B%2C%7D79" alt="x=0{,}79"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-1%7B%2C%7D4" alt="\cos x=-1{,}4"/></div> <div>ei ratkaisua</div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D-0%7B%2C%7D45" alt="\sin x=-0{,}45"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-0%7B%2C%7D47" alt="x=-0{,}47"/></div> 2019-11-19T11:43:51+02:00 https://peda.net/id/d18558dc0aa 2019-11-19T11:34:43+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=6%5Csin%20x%2B3%3D0" alt="6\sin x+3=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%5Csin%20x%3D-3" alt="6\sin x=-3"/><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%7B7%5Cpi%7D%7B6%7D" alt="x=\frac{7\pi}{6}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccos%20x%2B%5Csqrt%7B3%7D%3D0" alt="2\cos x+\sqrt{3}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos x=-\frac{\sqrt{3}}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D" alt="x=\frac{5\pi}{6}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%2B1%7B%2C%7D23%3D0" alt="\sin x+1{,}23=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D-1%7B%2C%7D23" alt="\sin x=-1{,}23"/></div> <div>siniä x ei ole määritelty kohdassa, ei ratkaisua</div> 2019-11-19T11:34:43+02:00 https://peda.net/id/1f8c05220aa 2019-11-19T11:29:45+02:00 a)<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" alt="x=\frac{\pi}{6}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D" alt="\cos x=\frac{1}{\sqrt{2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D" alt="x=\frac{\pi}{4}"/><br/> c)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Csin%5Cfrac%7B%5Cpi%7D%7B5%7D" alt="\sin x=\sin\frac{\pi}{5}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B5%7D" alt="x=\frac{\pi}{5}"/></div> 2019-11-19T11:29:45+02:00 https://peda.net/id/8e2f11140aa 2019-11-19T11:25:41+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D0%7B%2C%7D62%7B%2C%7D%5C%20x%5Capprox38%C2%B0" alt="\sin x=0{,}62{,}\ x\approx38°"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D0%7B%2C%7D33%7B%2C%7D%5C%20x%5Capprox71%C2%B0" alt="\cos x=0{,}33{,}\ x\approx71°"/></div> <div>c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Ccos54%C2%B0%7B%2C%7D%5C%20x%3D54%C2%B0" alt="\cos x=\cos54°{,}\ x=54°"/></div> 2019-11-19T11:25:41+02:00 https://peda.net/id/502b55540aa 2019-11-19T11:16:47+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Csin%5C%20%5Cfrac%7Bx%7D%7B3%7D%2B%5Csqrt%7B2%7D%3D0" alt="2\sin\ \frac{x}{3}+\sqrt{2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D" alt="\sin\ \frac{x}{3}=-\frac{\sqrt{2}}{2}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7Bx%7D%7B3%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D" alt="\sin\ \frac{x}{3}=-\frac{1}{\sqrt{2}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B5%5Cpi%7D%7B4%7D" alt="\frac{5\pi}{4}"/></div> <div>kaikki ratkaisut ovat:</div> <div><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"/><br/> <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"/></div> <div>tai</div> <div><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" alt="\frac{x}{3}=\pi-\frac{5\pi}{4}+n\cdot2\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot6%5Cpi" alt="x=-\frac{3\pi}{4}+n\cdot6\pi"/></div> <div> </div> <div>halutaan ratkaisut välillä ]-12π,12π[</div> <div>eli</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bmatrix%7D%0An%26x%3D%5Cfrac%7B15%5Cpi%7D%7B4%7D%2Bn%5Ccdot6%5Cpi%26x%3D-%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot6%5Cpi%5C%5C%0A0%26%5Cfrac%7B15%5Cpi%7D%7B4%7D%26-%5Cfrac%7B3%5Cpi%7D%7B4%7D%5C%5C%0A1%26%5Cfrac%7B39%5Cpi%7D%7B4%7D%26%5Cfrac%7B21%5Cpi%7D%7B4%7D%5C%5C%0A-1%26-%5Cfrac%7B9%5Cpi%7D%7B4%7D%26-%5Cfrac%7B27%5Cpi%7D%7B4%7D%5C%5C%0A2%26%5Cfrac%7B63%5Cpi%7D%7B4%7D%5Cleft(hyl.%5Cright)%26%5C%20%5Cfrac%7B45%5Cpi%7D%7B4%7D%5C%5C%0A-2%26-%5Cfrac%7B33%5Cpi%7D%7B4%7D%26x%3D-%5Cfrac%7B51%5Cpi%7D%7B4%7D%5C%20%5Cleft(hyl.%5Cright)%5C%5C%0A-3%26-%5Cfrac%7B57%5Cpi%7D%7B4%7D%5Cleft(hyl.%5Cright)%26%0A%5Cend%7Bmatrix%7D" alt="\begin{matrix}&#10;n&amp;x=\frac{15\pi}{4}+n\cdot6\pi&amp;x=-\frac{3\pi}{4}+n\cdot6\pi\\&#10;0&amp;\frac{15\pi}{4}&amp;-\frac{3\pi}{4}\\&#10;1&amp;\frac{39\pi}{4}&amp;\frac{21\pi}{4}\\&#10;-1&amp;-\frac{9\pi}{4}&amp;-\frac{27\pi}{4}\\&#10;2&amp;\frac{63\pi}{4}\left(hyl.\right)&amp;\ \frac{45\pi}{4}\\&#10;-2&amp;-\frac{33\pi}{4}&amp;x=-\frac{51\pi}{4}\ \left(hyl.\right)\\&#10;-3&amp;-\frac{57\pi}{4}\left(hyl.\right)&amp;&#10;\end{matrix}"/></div> <div>ratkaisuista halutulle välille kuuluvat <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B33%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20-%5Cfrac%7B27%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20-%5Cfrac%7B9%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20-%5Cfrac%7B3%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B15%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B21%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B35%5Cpi%7D%7B4%7D%5C%20tai%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{35\pi}{4}\ tai\ \frac{45\pi}{4}"/></div> <div> </div> 2019-11-19T11:16:47+02:00