Question

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

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:

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