https://peda.net/id/f7010e48052 2019-11-12T10:18:10+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/1a1df714053 2019-11-12T11:39:58+02:00 a) sin kuvastaa kehäpisteen y-koordinaattia, se on positiivinen ylemmillä neljänneksillä<br/> cos kuvastaa kehäpisteen x-koordinaattia, se on positiivinen oikeanpuoleisilla neljänneksillä<br/> b)<br/> α on tällöin välillä <img src="https://math-demo.abitti.fi/math.svg?latex=0%C2%B0-90%C2%B0" alt="0°-90°"/>tai<img src="https://math-demo.abitti.fi/math.svg?latex=0-%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="0-\frac{\pi}{2}"/><br/> c) <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E2%5Calpha%5Csin%5E2%5Calpha%3D1" alt="\cos^2\alpha\sin^2\alpha=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5E2%5Calpha%3D0%7B%2C%7D96%5E2" alt="\sin^2\alpha=0{,}96^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E2%5Calpha%3D1-0%7B%2C%7D96%5E2" alt="\cos^2\alpha=1-0{,}96^2"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E2%5Calpha%3D0%7B%2C%7D0784" alt="\cos^2\alpha=0{,}0784"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Calpha%3D%5Cpm0%7B%2C%7D28" alt="\cos\ \alpha=\pm0{,}28"/><br/> kulman α ollessa tylppä, cos α=-0,28<br/> kulman α ollessa terävä, cos α=0,28<br/> 2019-11-12T11:37:54+02:00 https://peda.net/id/b11fea84052 2019-11-12T11:27:48+02:00 a) <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin390%C2%B0%3D%5Csin30%C2%B0%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin390°=\sin30°=\frac{1}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cright)%3D%5Csin%5C%20%5Cfrac%7B7%5Cpi%7D%7B4%7D%3D%5Cfrac%7B-%5Csqrt%7B2%7D%7D%7B2%7D" alt="\sin\left(-\frac{\pi}{4}\right)=\sin\ \frac{7\pi}{4}=\frac{-\sqrt{2}}{2}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos2020%5Cpi%3D%5Ccos0%3D1" alt="\cos2020\pi=\cos0=1"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7B14%5Cpi%7D%7B3%7D%3D%5Csin%5C%20%5Cfrac%7B2%5Cpi%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\sin\ \frac{14\pi}{3}=\sin\ \frac{2\pi}{3}=\frac{\sqrt{3}}{2}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B11%5Cpi%7D%7B4%7D%3D%5Ccos%5C%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D%5Cfrac%7B-%5Csqrt%7B2%7D%7D%7B2%7D" alt="\cos\ \frac{11\pi}{4}=\cos\ \frac{3\pi}{4}=\frac{-\sqrt{2}}{2}"/></div> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B29%5Cpi%7D%7B6%7D%3D%5Cfrac%7B5%5Cpi%7D%7B6%7D%3D%5Cfrac%7B-%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\ \frac{29\pi}{6}=\frac{5\pi}{6}=\frac{-\sqrt{3}}{2}"/></div> <div> </div> 2019-11-12T11:27:48+02:00 https://peda.net/id/7bd8c612052 2019-11-12T11:19:09+02:00 a) 0,8<br/> b) 0,3<br/> c) 0<br/> d) -0,8<br/> e) 0<br/> f) 1 2019-11-12T11:19:09+02:00 https://peda.net/id/f8b376b0052 2019-11-12T11:15:29+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D" alt="\sin\ \frac{\pi}{4}=\frac{\sqrt{2}}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B7%5Cpi%7D%7B6%7D%3D%5Cfrac%7B-%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\ \frac{7\pi}{6}=\frac{-\sqrt{3}}{2}"/></div> <div>c) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7B8%5Cpi%7D%7B3%7D%3D%5Csin%5C%20%5Cfrac%7B2%5Cpi%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\sin\ \frac{8\pi}{3}=\sin\ \frac{2\pi}{3}=\frac{\sqrt{3}}{2}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B9%5Cpi%7D%7B4%7D%3D%5Ccos%5C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D" alt="\cos\ \frac{9\pi}{4}=\cos\ \frac{\pi}{4}=\frac{\sqrt{2}}{2}"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright)%3D%5Csin%5C%20%5Cfrac%7B5%5Cpi%7D%7B3%7D%3D%5Cfrac%7B-%5Csqrt%7B3%7D%7D%7B2%7D" alt="\sin\left(-\frac{\pi}{3}\right)=\sin\ \frac{5\pi}{3}=\frac{-\sqrt{3}}{2}"/></div> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D%5Ccos%5C%20%5Cfrac%7B11%5Cpi%7D%7B6%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\left(-\frac{\pi}{6}\right)=\cos\ \frac{11\pi}{6}=\frac{\sqrt{3}}{2}"/></div> <div> </div> 2019-11-12T11:15:29+02:00 https://peda.net/id/f6563bf6052 2019-11-12T11:08:16+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7B%5Cpi%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin\ \frac{\pi}{6}=\frac{1}{2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B%5Cpi%7D%7B6%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\ \frac{\pi}{6}=\frac{\sqrt{3}}{2}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D" alt="\sin\ \frac{3\pi}{4}=\frac{1}{\sqrt{2}}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Cfrac%7B3%5Cpi%7D%7B4%7D%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D" alt="\cos\ \frac{3\pi}{4}=-\frac{1}{\sqrt{2}}"/></div> 2019-11-12T11:08:16+02:00 https://peda.net/id/186c2860052 2019-11-12T10:51:00+02:00 Olkoon P = (x, y) suunnattua kulmaa α vastaava kehäpiste yksikköympyrällä.<br/> Kulman α<br/> a)<br/> kosini on kehäpisteen x-koordinaatti<br/> b)<br/> sini on kehäpisteen y-koordinaatti<br/> Huom: Kehäpiste P =(x, y) = (cos α, sin α)<br/> Huom: -1 ≤ cos α ≤ 1 sekä -1 ≤ sin α ≤ 1<br/> <div>- Kulman kehäpiste ei muutu, jos kulmaan lisätään tai vähennetään täysiä kulmia 2π.</div> <div> </div> <div>Lause</div> <div>Kaikille n∈ℤ pätee </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5C%20%5Calpha%3D%5Csin%5Cleft(%5Calpha%2Bn%5Ccdot2%5Cpi%5Cright)" alt="\sin\ \alpha=\sin\left(\alpha+n\cdot2\pi\right)"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5C%20%5Calpha%3D%5Ccos%5Cleft(%5Calpha%2Bn%5Ccdot2%5Cpi%5Cright)" alt="\cos\ \alpha=\cos\left(\alpha+n\cdot2\pi\right)"/><br/> <br/> <br/> </div> <div> <div>esim</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin30%C2%B0%3D%5Cfrac%7B1%7D%7B2%7D" alt="\sin30°=\frac{1}{2}"/></div> <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%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%2B%5Cfrac%7B12%5Cpi%7D%7B6%7D%5Cright)%3D%5Ccos%5Cleft(%5Cfrac%7B11%5Cpi%7D%7B6%7D%5Cright)%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\left(-\frac{\pi}{6}\right)=\cos\left(-\frac{\pi}{6}+2\pi\right)=\cos\left(-\frac{\pi}{6}+\frac{12\pi}{6}\right)=\cos\left(\frac{11\pi}{6}\right)=\frac{\sqrt{3}}{2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Cleft(%5Cfrac%7B17%7D%7B6%7D%5Cpi%5Cright)%3D%5Ccos%5Cleft(%5Cfrac%7B5%7D%7B6%7D%5Cpi%5Cright)%3D%5Cfrac%7B-%5Csqrt%7B3%7D%7D%7B2%7D" alt="\cos\left(\frac{17}{6}\pi\right)=\cos\left(\frac{5}{6}\pi\right)=\frac{-\sqrt{3}}{2}"/></div> </div> 2019-11-12T10:19:06+02:00