use the given information to find the exact value of a. sin 2θ, b. cos 2θ, and c. tan 2θ. cosθ = 12/20, θ…

Explanation:

Step1: Find $\sin\theta$

Use the identity $\sin^{2}\theta+\cos^{2}\theta = 1$. Since $\cos\theta=\frac{12}{20}=\frac{3}{5}$ and $\theta$ is in quadrant IV (where $\sin\theta<0$), we have: $$\sin\theta=-\sqrt{1 - \cos^{2}\theta}=-\sqrt{1-\left(\frac{3}{5}\right)^{2}}=-\sqrt{1-\frac{9}{25}}=-\sqrt{\frac{16}{25}}=-\frac{4}{5}$$

Step2: Calculate $\sin2\theta$

Use the double - angle formula $\sin2\theta = 2\sin\theta\cos\theta$. Substitute $\sin\theta=-\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$: $$\sin2\theta=2\times\left(-\frac{4}{5}\right)\times\frac{3}{5}=-\frac{24}{25}$$

Step3: Calculate $\cos2\theta$

Use the double - angle formula $\cos2\theta=\cos^{2}\theta-\sin^{2}\theta$. Substitute $\sin\theta=-\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$: $$\cos2\theta=\left(\frac{3}{5}\right)^{2}-\left(-\frac{4}{5}\right)^{2}=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25}$$

Step4: Calculate $\tan2\theta$

Use the formula $\tan2\theta=\frac{\sin2\theta}{\cos2\theta}$. Substitute $\sin2\theta=-\frac{24}{25}$ and $\cos2\theta=-\frac{7}{25}$: $$\tan2\theta=\frac{-\frac{24}{25}}{-\frac{7}{25}}=\frac{24}{7}$$

Answer:

a. $\sin2\theta =-\frac{24}{25}$ b. $\cos2\theta=-\frac{7}{25}$ c. $\tan2\theta=\frac{24}{7}$