Question

Use identities to find values of the sine and cosine functions of the function for the angle measure
a. theta, given that cos2theta=28/53 and 0theta < theta < 90degrees
b. 2theta, given sin theta= - sqrt 7 over 5 and cos theta > 0
c. 2x, given tan x=2 and cos x<0

Answer 1

Answer:

Step-by-step explanation:

a) Given cos2theta=28/53 and 0degrees< theta < 90degrees

From cos2theta=28/53

$2\theta = cos^{-1}\frac{28}{53}$

$2\theta = cos^{-1}0.5283\\ \\2\theta = 58.12\\\\Dividing\ both \ sides\ by \ 2\\\\\frac{2\theta}{2} = \frac{58.12}{2}\\ \\\theta = 29.06^0$

b) Given

$sin\theta = \frac{-\sqrt{7} }{5} \\\\\theta = sin^{-1} \frac{-\sqrt{7} }{5}\\\\\\\theta = sin^{-1} \frac{-2.6458}{5}\\\\\theta = sin^{-1} -0.5292\\\\\theta = -31.95^0$

If cos theta $\gneq$ 0, this means we need to look for the quadrant where sin is negative and cos is positive. That will be the fourth quadrant. In the fourth quadrant, theta = 360 - 31.95° = 328.05°

2theta = 2 * 328.05

2theta = 656.1°

c) Given tan x=2 and cos x<0, lets find the angle of x first.

If tan x = 2

x = tan^-1 2

x = 63.4°

Sine cos is less than 0, then we need to find the angle of x where tan is positive and cos is negative. That will be the third quadrant. In the third quadrant, ew value of x = 180+63.4

x = 243.4°

Since we are to find 2x,

2x = 2(243.4)

2x = 486.8°