Question

use the given information to find the exact value of a. sin 2θ, b. cos 2θ, and c. tan 2θ. cos θ = 24/26, θ lies in quadrant iv a. sin 2θ = -120/169 (type an integer or a fraction. simplify your answer.) b. cos 2θ = (type an integer or a fraction. simplify your answer.)

Explanation:

Step1: Find \(\sin\theta\)

Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), we have \(\sin\theta=-\sqrt{1-\cos^{2}\theta}\) (since \(\theta\) is in quadrant IV, \(\sin\theta<0\)).
\(\cos\theta=\frac{24}{26}=\frac{12}{13}\), so \(\sin\theta=-\sqrt{1 - (\frac{12}{13})^{2}}=-\sqrt{\frac{169 - 144}{169}}=-\sqrt{\frac{25}{169}}=-\frac{5}{13}\)

Step2: Calculate \(\cos2\theta\) using the double - angle formula \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\)

Substitute \(\cos\theta=\frac{12}{13}\) and \(\sin\theta =-\frac{5}{13}\) into the formula:
\(\cos2\theta=(\frac{12}{13})^{2}-(-\frac{5}{13})^{2}=\frac{144}{169}-\frac{25}{169}=\frac{144 - 25}{169}=\frac{119}{169}\)

Answer:

\(\frac{119}{169}\)