https://peda.net/id/1574efac15a 2019-12-03T10:22:45+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/6ce0b9a8203 2019-12-16T22:05:36+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin3x-%5Csqrt%7B3%7D%5Ccos3x%3D0" alt="\sin3x-\sqrt{3}\cos3x=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin3x%7D%7B%5Ccos3x%7D%3D%5Ctan3x" alt="\frac{\sin3x}{\cos3x}=\tan3x"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin3x-%5Csqrt%7B3%7D%5Ccos3x%3D0" alt="\sin3x-\sqrt{3}\cos3x=0"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin3x%3D%5Csqrt%7B3%7D%5Ccdot%5Ccos3x%5Cparallel%3A%5Ccos3x" alt="\sin3x=\sqrt{3}\cdot\cos3x\parallel:\cos3x"/>, cos x ei ole nolla, sillä yhtälö ei toteudu, jos cos x on nolla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin3x%7D%7B%5Ccos3x%7D%3D%5Csqrt%7B3%7D" alt="\frac{\sin3x}{\cos3x}=\sqrt{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan3x%3D%5Csqrt%7B3%7D" alt="\tan3x=\sqrt{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%2Bn%5Ccdot%5Cpi%5Cparallel%3A3" alt="3x=\frac{\pi}{3}+n\cdot\pi\parallel:3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B9%7D%2Bn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=\frac{\pi}{9}+n\cdot\frac{\pi}{3}"/></div> 2019-12-16T22:05:36+02:00 https://peda.net/id/b7f4b722167 2019-12-04T10:47:35+02:00 <div>a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D3" alt="\tan x=3"/><br/> yksi ratkaisu<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5E%7B-1%7D%5Cleft(3%5Cright)%3Dx%3D1%7B%2C%7D24904...%5Capprox1%7B%2C%7D2" alt="\tan^{-1}\left(3\right)=x=1{,}24904...\approx1{,}2"/><br/> kaikki ratkaisut<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%7B%2C%7D2%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=1{,}2+n\cdot\pi{,}\ n\in\mathbb{Z}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3Dn" alt="\tan x=n"/>ratkaisuja saadaan suoran <img src="https://math-demo.abitti.fi/math.svg?latex=y%3Dn%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BR%7D" alt="y=n{,}\ n\in\mathbb{R}"/> ja kuvaajan leikkauspisteistä <br/> <br/> </div> 2019-12-04T10:47:35+02:00 https://peda.net/id/204cc1a0167 2019-12-04T10:29:02+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D" alt="\tan x=-\frac{1}{\sqrt{3}}"/><br/> yksi ratkaisu<br/> <div><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>kaikki ratkaisut</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{5\pi}{6}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div> </div> <div>ratkaisu välillä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7B%5D%7D-%5Cfrac%7B3%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%5Ctext%7B%5B%7D" alt="\text{]}-\frac{3\pi}{2}{,}\ -\frac{\pi}{2}\text{[}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n%3D-2" alt="n=-2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D%2B%5Cleft(-2%5Cright)%5Ccdot%5Cpi%3D-%5Cfrac%7B7%5Cpi%7D%7B6%7D" alt="x=\frac{5\pi}{6}+\left(-2\right)\cdot\pi=-\frac{7\pi}{6}"/><br/> <br/> </div> <div>ratkaisu välillä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7B%5D%7D%5Cfrac%7B5%5Cpi%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B7%5Cpi%7D%7B2%7D%5Ctext%7B%5B%7D" alt="\text{]}\frac{5\pi}{2}{,}\ \frac{7\pi}{2}\text{[}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n%3D2" alt="n=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D%2B2%5Ccdot%5Cpi%3D%5Cfrac%7B17%5Cpi%7D%7B6%7D" alt="x=\frac{5\pi}{6}+2\cdot\pi=\frac{17\pi}{6}"/></div> 2019-12-04T10:29:02+02:00 https://peda.net/id/a8cda19415b 2019-12-03T11:45:37+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D2%5Ctan%20x-%5Csin2x" alt="f\left(x\right)=2\tan x-\sin2x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(0%5Cright)%3D2%5Ctan0-%5Csin2%5Ccdot0%3D2%5Ccdot0-0%3D0" alt="f\left(0\right)=2\tan0-\sin2\cdot0=2\cdot0-0=0"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%3D2%5Ctan%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Csin%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="f\left(\frac{\pi}{4}\right)=2\tan\frac{\pi}{4}-\sin\frac{\pi}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2-1%3D1" alt="2-1=1"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D2%5Ctan%5Cfrac%7B%5Cpi%7D%7B6%7D-%5Csin%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="f\left(\frac{\pi}{6}\right)=2\tan\frac{\pi}{6}-\sin\frac{\pi}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Ccdot%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%3D%5Cfrac%7B2%7D%7B%5Csqrt%7B3%7D%7D-%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="2\cdot\frac{1}{\sqrt{3}}-\frac{\sqrt{3}}{2}=\frac{2}{\sqrt{3}}-\frac{\sqrt{3}}{2}"/></div> 2019-12-03T11:45:37+02:00 https://peda.net/id/ad12258415a 2019-12-03T11:30:19+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan3x%3D%5Ctan39%C2%B0" alt="\tan3x=\tan39°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D39%C2%B0%2Bn%5Ccdot180%C2%B0" alt="3x=39°+n\cdot180°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D13%C2%B0%2Bn%5Ccdot60%C2%B0%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=13°+n\cdot60°{,}\ n\in\mathbb{Z}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan2x%3D-4" alt="\tan2x=-4"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D%5Ctan%5E%7B-1%7D%5Cleft(-4%5Cright)%2Bn%5Ccdot180%C2%B0" alt="2x=\tan^{-1}\left(-4\right)+n\cdot180°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-38%C2%B0%2Bn%5Ccdot90%C2%B0%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=-38°+n\cdot90°{,}\ n\in\mathbb{Z}"/><br/> <br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cfrac%7Bx%7D%7B2%7D%3D1" alt="\tan\frac{x}{2}=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bx%7D%7B2%7D%3D%5Ctan%5E%7B-1%7D%5Cleft(1%5Cright)%2Bn%5Ccdot180%C2%B0" alt="\frac{x}{2}=\tan^{-1}\left(1\right)+n\cdot180°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D90%C2%B0%2Bn%5Ccdot360%C2%B0%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=90°+n\cdot360°{,}\ n\in\mathbb{Z}"/> 2019-12-03T11:24:16+02:00 https://peda.net/id/cb9af34215a 2019-12-03T11:17:57+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D1" alt="\tan x=1"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{\pi}{4}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D-1" alt="\tan x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B3%5Cpi%7D%7B4%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{3\pi}{4}+n\cdot\pi{,}\ n\in\mathbb{Z}"/><br/> c)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D0" alt="\tan x=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3Dn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> 2019-12-03T11:17:57+02:00 https://peda.net/id/42c89dd015a 2019-12-03T11:14:08+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D" alt="\tan x=\frac{1}{\sqrt{3}}"/><br/> <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}"/></div> <div>kaikki ratkaisut</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B6%7D%2Bn%5Ccdot%5Cpi%7B%2C%7D%5C%20n%5Cin%5Cmathbb%7BZ%7D" alt="x=\frac{\pi}{6}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan3x%3D%5Ctan%5Cfrac%7B6%5Cpi%7D%7B7%7D" alt="\tan3x=\tan\frac{6\pi}{7}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=3x%3D%5Cfrac%7B6%5Cpi%7D%7B7%7D%2Bn%5Ccdot%5Cpi" alt="3x=\frac{6\pi}{7}+n\cdot\pi"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B6%5Cpi%7D%7B21%7D%2Bn%5Ccdot%5Cfrac%7B%5Cpi%7D%7B3%7D" alt="x=\frac{6\pi}{21}+n\cdot\frac{\pi}{3}"/></div> 2019-12-03T11:14:08+02:00 https://peda.net/id/bba67b4c15a 2019-12-03T11:10:21+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Csqrt%7B3%7D%7B%2C%7D%5C%20x%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B4%5Cpi%7D%7B3%7D%7B%2C%7D%5C%20%5Cfrac%7B7%5Cpi%7D%7B3%7D%7B%2C%7D%5Cfrac%7B11%5Cpi%7D%7B3%7D" alt="\tan x=\sqrt{3}{,}\ x=\frac{\pi}{3}{,}\ \frac{4\pi}{3}{,}\ \frac{7\pi}{3}{,}\frac{11\pi}{3}"/></div> <div> </div> 2019-12-03T11:10:21+02:00 https://peda.net/id/33ca876815a 2019-12-03T11:05:17+02:00 <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>-arvojoukko on -]-∞,∞[ eli ℝ</div> <div>-jatkuva määrittelyjoukossaan</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>-funktio on jaksollinen, perusjakso π</div> <div>-funktio on kasvava kaikilla määrittelyjoukkonsa osaväleillä</div> <div> </div> <div>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}"/><br/> <br/> <br/> <br/> <div>tangentin derivointikaava</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%7BZ%7D" alt="D\tan x=\frac{1}{\cos^2x}=1+\tan^2x{,}\ kun\ x\ne\frac{\pi}{2}+n\cdot\pi{,}\ n\in\mathbb{Z}"/></div> </div> 2019-12-03T10:30:46+02:00