Question

use the given information to find the exact value of a. sin 20, b. cos
cos 0 = \frac { 20 } { 20 }, 0 lies in quadrant iv
a. \sin 2 \theta = \square (type an integer or a fraction. simplify your
b. \cos 2 \theta = \square (type an integer or a fraction. simplify
c. \tan 2 \theta = \square (type an integer or a fraction. simpl

Explanation:

Step1: Find \(\sin\theta\)

Use the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\). Given \(\cos\theta=\frac{20}{29}\), then \(\sin^{2}\theta=1-\cos^{2}\theta=1 - (\frac{20}{29})^{2}=\frac{841 - 400}{841}=\frac{441}{841}\). Since \(\theta\) is in quadrant IV, \(\sin\theta=-\frac{21}{29}\).

Step2: Calculate \(\sin2\theta\)

Use the double - angle formula \(\sin2\theta = 2\sin\theta\cos\theta\). Substitute \(\sin\theta=-\frac{21}{29}\) and \(\cos\theta=\frac{20}{29}\), we get \(\sin2\theta=2\times(-\frac{21}{29})\times\frac{20}{29}=-\frac{840}{841}\).

Step3: Calculate \(\cos2\theta\)

Use the double - angle formula \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\). Substitute \(\sin\theta=-\frac{21}{29}\) and \(\cos\theta=\frac{20}{29}\), we get \(\cos2\theta=(\frac{20}{29})^{2}-(-\frac{21}{29})^{2}=\frac{400 - 441}{841}=-\frac{41}{841}\).

Step4: Calculate \(\tan2\theta\)

Use the formula \(\tan2\theta=\frac{\sin2\theta}{\cos2\theta}\). Substitute \(\sin2\theta=-\frac{840}{841}\) and \(\cos2\theta=-\frac{41}{841}\), we get \(\tan2\theta=\frac{-\frac{840}{841}}{-\frac{41}{841}}=\frac{840}{41}\).

Answer:

a. \(-\frac{840}{841}\)
b. \(-\frac{41}{841}\)
c. \(\frac{840}{41}\)