https://peda.net/id/d581bda8129 2019-11-29T12:41:43+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/a6ccede012a 2019-11-29T14:13:28+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-2%5Cpi%7B%2C%7D%5C%20-%5Cpi%7B%2C%7D%5C%200%7B%2C%7D%5C%20%5Cpi%7B%2C%7D%5C%202%5Cpi" alt="-2\pi{,}\ -\pi{,}\ 0{,}\ \pi{,}\ 2\pi"/><br/> b)<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" alt="x=\frac{\pi}{4}+n\cdot\pi"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B7%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20-%5Cfrac%7B3%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B5%5Cpi%7D%7B4%7D%7B%2C%7D%5C%20%5Cfrac%7B9%5Cpi%7D%7B4%7D" alt="-\frac{7\pi}{4}{,}\ -\frac{3\pi}{4}{,}\ \frac{\pi}{4}{,}\ \frac{5\pi}{4}{,}\ \frac{9\pi}{4}"/></div> 2019-11-29T14:13:28+02:00 https://peda.net/id/4f3e654a12a 2019-11-29T14:11:01+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D-3" alt="\tan x=\frac{\sin x}{\cos x}=-3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3C0" alt="\sin x&lt;0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B-0%7B%2C%7D9%7D%7B0%7B%2C%7D3%7D" alt="\tan x=\frac{-0{,}9}{0{,}3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D-0%7B%2C%7D9" alt="\sin x=-0{,}9"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D0%7B%2C%7D3" alt="\cos x=0{,}3"/> 2019-11-29T14:11:01+02:00 https://peda.net/id/32eeb97c12a 2019-11-29T14:03:04+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cfrac%7B%5Cpi%7D%7B6%7D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D" alt="\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(%5Cfrac%7B%5Cpi%7D%7B6%7D-%5Cpi%5Cright)%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D" alt="\tan\left(\frac{\pi}{6}-\pi\right)=-\frac{1}{\sqrt{3}}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(%5Cfrac%7B%5Cpi%7D%7B3%7D%2B3%5Cpi%5Cright)%3D%5Csqrt%7B3%7D" alt="\tan\left(\frac{\pi}{3}+3\pi\right)=\sqrt{3}"/></div> 2019-11-29T14:03:04+02:00 https://peda.net/id/4e596348129 2019-11-29T13:56:41+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D%5Cfrac%7B1%7D%7B4%7D" alt="\cos x=\frac{1}{4}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E2x%2B%5Csin%5E2x%3D1" alt="\cos^2x+\sin^2x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cpm%5Csqrt%7B1-%5Cfrac%7B1%7D%7B16%7D%7D" alt="\sin x=\pm\sqrt{1-\frac{1}{16}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D%5Cpm0%7B%2C%7D9682...%5Capprox%5Cpm0%7B%2C%7D97" alt="\sin x=\pm0{,}9682...\approx\pm0{,}97"/></div> <div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B%5Cpm0%7B%2C%7D9682...%7D%7B%5Cfrac%7B1%7D%7B4%7D%7D%3D%5Cpm3%7B%2C%7D872...%5Capprox3%7B%2C%7D87" alt="\tan x=\frac{\sin x}{\cos x}=\frac{\pm0{,}9682...}{\frac{1}{4}}=\pm3{,}872...\approx3{,}87"/></div> 2019-11-29T13:56:41+02:00 https://peda.net/id/d290506e129 2019-11-29T13:53:13+02:00 tangentti kasvaa ensin nollasta kunnes kohdassa <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D" alt="x=\frac{\pi}{2}"/> sitä ei ole määritelty, sen jälkeen tangentti lähestyy nollaa negatiiviselta puolelta 2019-11-29T13:53:13+02:00 https://peda.net/id/798aae60129 2019-11-29T13:50:44+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B-%5Cfrac%7B5%7D%7B13%7D%7D%7B%5Cfrac%7B12%7D%7B13%7D%7D%3D-%5Cfrac%7B5%7D%7B12%7D" alt="\tan x=\frac{\sin x}{\cos x}=\frac{-\frac{5}{13}}{\frac{12}{13}}=-\frac{5}{12}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-0%7B%2C%7D6%3B0%7B%2C%7D8%5Cright)" alt="\left(-0{,}6;0{,}8\right)"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-0%7B%2C%7D6" alt="\cos x=-0{,}6"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D0%7B%2C%7D8" alt="\sin x=0{,}8"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B0%7B%2C%7D8%7D%7B-0%7B%2C%7D6%7D%3D-%5Cfrac%7B4%7D%7B3%7D" alt="\tan x=\frac{0{,}8}{-0{,}6}=-\frac{4}{3}"/></div> 2019-11-29T13:50:44+02:00 https://peda.net/id/9bb64b1c129 2019-11-29T13:44:31+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cpi%7D%7B2%7D%7B%2C%7D%5Cfrac%7B3%5Cpi%7D%7B2%7D%7B%2C%7D%5Cfrac%7B5%5Cpi%7D%7B2%7D%7B%2C%7D%5Cfrac%7B7%5Cpi%7D%7B2%7D%7B%2C%7D%5Cfrac%7B9%5Cpi%7D%7B2%7D" alt="\frac{\pi}{2}{,}\frac{3\pi}{2}{,}\frac{5\pi}{2}{,}\frac{7\pi}{2}{,}\frac{9\pi}{2}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Ctan%5Cfrac%7B3%5Cpi%7D%7B4%7D%2B%5Ctan%5Cfrac%7B11%5Cpi%7D%7B12%7D%3D1-%5Cleft(-1%5Cright)%2B%5Cleft(-%5Cleft(2%2B%5Csqrt%7B3%7D%5Cright)%5Cright)%3D2-2-%5Csqrt%7B3%7D%3D-%5Csqrt%7B3%7D" alt="\tan\frac{\pi}{4}-\tan\frac{3\pi}{4}+\tan\frac{11\pi}{12}=1-\left(-1\right)+\left(-\left(2+\sqrt{3}\right)\right)=2-2-\sqrt{3}=-\sqrt{3}"/> 2019-11-29T13:44:31+02:00 https://peda.net/id/02d67d22129 2019-11-29T13:40:15+02:00 a) 117°<br/> b) 297° ja -63° 2019-11-29T13:40:15+02:00 https://peda.net/id/826e584e129 2019-11-29T13:36:40+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan50%C2%B0%3D1%7B%2C%7D2" alt="\tan50°=1{,}2"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan30%C2%B0%3D0%7B%2C%7D6" alt="\tan30°=0{,}6"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan230%C2%B0%3D1%7B%2C%7D2" alt="\tan230°=1{,}2"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(-30%C2%B0%5Cright)%3D-0%7B%2C%7D6" alt="\tan\left(-30°\right)=-0{,}6"/></div> <div>e)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(-150%C2%B0%5Cright)%3D0%7B%2C%7D6" alt="\tan\left(-150°\right)=0{,}6"/></div> <div>f)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(-130%C2%B0%5Cright)%3D1%7B%2C%7D2" alt="\tan\left(-130°\right)=1{,}2"/></div> 2019-11-29T13:36:40+02:00 https://peda.net/id/831b3aa8129 2019-11-29T13:15:12+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5C%20%5Cfrac%7B11%5Cpi%7D%7B3%7D%3D-%5Csqrt%7B3%7D" alt="\tan\ \frac{11\pi}{3}=-\sqrt{3}"/><br/> b)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(-%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cright)%3D-%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D" alt="\tan\left(-\frac{\pi}{6}\right)=-\frac{1}{\sqrt{3}}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Cleft(-%5Cfrac%7B9%5Cpi%7D%7B4%7D%5Cright)%3D-1" alt="\tan\left(-\frac{9\pi}{4}\right)=-1"/></div> 2019-11-29T13:15:12+02:00 https://peda.net/id/227ca57e129 2019-11-29T13:12:30+02:00 <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%20x%3D0%7B%2C%7D28%7B%2C%7D%5C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%3Cx%3C%5Cpi" alt="\sin x=0{,}28{,}\ \frac{\pi}{2}&lt;x&lt;\pi"/></div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5E2x%2B%5Ccos%5E2x%3D1" alt="\sin^2x+\cos^2x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5E2x%3D1-%5Csin%5E2x" alt="\cos^2x=1-\sin^2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Ccos%20x%5Cright)%5E2%3D1-0%7B%2C%7D28%5E2" alt="\left(\cos x\right)^2=1-0{,}28^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%20x%3D-0%7B%2C%7D96" alt="\cos x=-0{,}96"/></div> <div>negatiivinen koska <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cpi%7D%7B2%7D%3Cx%3C%5Cpi" alt="\frac{\pi}{2}&lt;x&lt;\pi"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%20x%3D%5Cfrac%7B%5Csin%20x%7D%7B%5Ccos%20x%7D%3D%5Cfrac%7B0%7B%2C%7D28%7D%7B-0%7B%2C%7D96%7D%3D-0%7B%2C%7D3038...%5Capprox-0%7B%2C%7D30" alt="\tan x=\frac{\sin x}{\cos x}=\frac{0{,}28}{-0{,}96}=-0{,}3038...\approx-0{,}30"/> 2019-11-29T13:12:30+02:00