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{12}{13} ) and θ terminates in quadrant i
( sin \theta=\frac{sqrt{26}}{26} )
(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 can use \(\cos2\theta=2\cos^{2}\theta - 1\) to find \(\cos\theta\).
Given \(\cos2\theta=\frac{12}{13}\), then \(2\cos^{2}\theta-1 = \frac{12}{13}\).

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

Add \(1\) to both sides of the equation \(2\cos^{2}\theta-1=\frac{12}{13}\):
\(2\cos^{2}\theta=\frac{12}{13}+ 1=\frac{12 + 13}{13}=\frac{25}{13}\).
Divide both sides by \(2\): \(\cos^{2}\theta=\frac{25}{26}\).

Step3: Find \(\cos\theta\)

Since \(\theta\) terminates in quadrant I, \(\cos\theta>0\). So \(\cos\theta=\sqrt{\frac{25}{26}}=\frac{5}{\sqrt{26}}=\frac{5\sqrt{26}}{26}\).

Answer:

\(\cos\theta=\frac{5\sqrt{26}}{26}\)