3.1 Tangentti yksikköympyrässä

314

a)
$-2\pi{,}\ -\pi{,}\ 0{,}\ \pi{,}\ 2\pi$
b)
$\tan x=1$

$x=\frac{\pi}{4}+n\cdot\pi$

$-\frac{7\pi}{4}{,}\ -\frac{3\pi}{4}{,}\ \frac{\pi}{4}{,}\ \frac{5\pi}{4}{,}\ \frac{9\pi}{4}$

309

$\tan x=\frac{\sin x}{\cos x}=-3$
$\sin x<0$
$\tan x=\frac{-0{,}9}{0{,}3}$
$\sin x=-0{,}9$
$\cos x=0{,}3$

312

a)

$\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}$

b)

$\tan\left(\frac{\pi}{6}-\pi\right)=-\frac{1}{\sqrt{3}}$

c)

$\tan\left(\frac{\pi}{3}+3\pi\right)=\sqrt{3}$

306

$\cos x=\frac{1}{4}$
$\cos^2x+\sin^2x=1$
$\sin x=\pm\sqrt{1-\frac{1}{16}}$

$\sin x=\pm0{,}9682...\approx\pm0{,}97$

$\tan x=\frac{\sin x}{\cos x}=\frac{\pm0{,}9682...}{\frac{1}{4}}=\pm3{,}872...\approx3{,}87$

305

tangentti kasvaa ensin nollasta kunnes kohdassa $x=\frac{\pi}{2}$ sitä ei ole määritelty, sen jälkeen tangentti lähestyy nollaa negatiiviselta puolelta

304

a)
$\tan x=\frac{\sin x}{\cos x}=\frac{-\frac{5}{13}}{\frac{12}{13}}=-\frac{5}{12}$
b)
$\left(-0{,}6;0{,}8\right)$

$\cos x=-0{,}6$
$\sin x=0{,}8$

$\tan x=\frac{0{,}8}{-0{,}6}=-\frac{4}{3}$

303

a)
$\frac{\pi}{2}{,}\frac{3\pi}{2}{,}\frac{5\pi}{2}{,}\frac{7\pi}{2}{,}\frac{9\pi}{2}$
b)
$\tan\frac{\pi}{4}-\tan\frac{3\pi}{4}+\tan\frac{11\pi}{12}=1-\left(-1\right)+\left(-\left(2+\sqrt{3}\right)\right)=2-2-\sqrt{3}=-\sqrt{3}$

302

a) 117°
b) 297° ja -63°

301

a)

$\tan50°=1{,}2$

b)

$\tan30°=0{,}6$

c)

$\tan230°=1{,}2$

d)

$\tan\left(-30°\right)=-0{,}6$

e)

$\tan\left(-150°\right)=0{,}6$

f)

$\tan\left(-130°\right)=1{,}2$

313

a)
$\tan\ \frac{11\pi}{3}=-\sqrt{3}$
b)

$\tan\left(-\frac{\pi}{6}\right)=-\frac{1}{\sqrt{3}}$

c)

$\tan\left(-\frac{9\pi}{4}\right)=-1$

307

$\sin x=0{,}28{,}\ \frac{\pi}{2}<x<\pi$

$\sin^2x+\cos^2x=1$
$\cos^2x=1-\sin^2x$
$\left(\cos x\right)^2=1-0{,}28^2$

$\cos x=-0{,}96$

negatiivinen koska $\frac{\pi}{2}<x<\pi$

$\tan x=\frac{\sin x}{\cos x}=\frac{0{,}28}{-0{,}96}=-0{,}3038...\approx-0{,}30$