use identities to find the values of the sine and cosine functions for the following angle measure. \nθ…

use identities to find the values of the sine and cosine functions for the following angle measure. \nθ, given that ( cos 2 \theta=\frac{28}{53} ) and θ terminates in quadrant i\n( sin \theta=\frac{5 sqrt{106}}{106} )\n(type an exact answer, using radicals as needed. rationalize all denominators.)\n( cos \theta=square )\n(type an exact answer, using radicals as needed. rationalize all denominators.)

use identities to find the values of the sine and cosine functions for the following angle measure. \nθ, given that ( cos 2 \theta=\frac{28}{53} ) and θ terminates in quadrant i\n( sin \theta=\frac{5 sqrt{106}}{106} )\n(type an exact answer, using radicals as needed. rationalize all denominators.)\n( cos \theta=square )\n(type an exact answer, using radicals as needed. rationalize all denominators.)

Answer

Explanation:

Step1: Use the double - angle identity for cosine

The double - angle identity for cosine is (\cos2\theta = 1 - 2\sin^{2}\theta=2\cos^{2}\theta - 1). We will use (\cos2\theta=2\cos^{2}\theta - 1) to find (\cos\theta). Given (\cos2\theta=\frac{28}{53}), then (2\cos^{2}\theta-1 = \frac{28}{53}).

Step2: Solve for (\cos^{2}\theta)

Add (1) to both sides of the equation (2\cos^{2}\theta-1=\frac{28}{53}): (2\cos^{2}\theta=\frac{28}{53}+1=\frac{28 + 53}{53}=\frac{81}{53}). Divide both sides by (2): (\cos^{2}\theta=\frac{81}{106}).

Step3: Solve for (\cos\theta)

Since (\theta) terminates in quadrant (I), (\cos\theta>0). So (\cos\theta=\sqrt{\frac{81}{106}}=\frac{9}{\sqrt{106}}). Rationalize the denominator: (\cos\theta=\frac{9\sqrt{106}}{106}).

Answer:

(\cos\theta=\frac{9\sqrt{106}}{106})