QuestionJune 18, 2025

Find the exact value of tan[2sin^-1(-(3)/(5))] tan[2sin^-1(-(3)/(5))]=square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Find the exact value of tan[2sin^-1(-(3)/(5))] tan[2sin^-1(-(3)/(5))]=square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Find the exact value of tan[2sin^-1(-(3)/(5))] tan[2sin^-1(-(3)/(5))]=square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution
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Answer

-\frac{24}{7} Explanation 1. Define the angle Let \theta = \sin^{-1}(-\frac{3}{5}). Then, \sin(\theta) = -\frac{3}{5}. 2. Find \cos(\theta) Use **Pythagorean identity**: \cos^2(\theta) = 1 - \sin^2(\theta). So, \cos^2(\theta) = 1 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25}. Thus, \cos(\theta) = \frac{4}{5} (since \theta is in the fourth quadrant). 3. Calculate \tan(2\theta) Use **double angle formula**: \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}. First, find \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}. 4. Apply the double angle formula Substitute into the formula: \tan(2\theta) = \frac{2(-\frac{3}{4})}{1-(-\frac{3}{4})^2} = \frac{-\frac{6}{4}}{1-\frac{9}{16}} = \frac{-\frac{3}{2}}{\frac{7}{16}} = -\frac{3}{2} \times \frac{16}{7} = -\frac{24}{7}.

Explanation

1. Define the angle<br /> Let $\theta = \sin^{-1}(-\frac{3}{5})$. Then, $\sin(\theta) = -\frac{3}{5}$.<br /><br />2. Find $\cos(\theta)$<br /> Use **Pythagorean identity**: $\cos^2(\theta) = 1 - \sin^2(\theta)$. So, $\cos^2(\theta) = 1 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25}$. Thus, $\cos(\theta) = \frac{4}{5}$ (since $\theta$ is in the fourth quadrant).<br /><br />3. Calculate $\tan(2\theta)$<br /> Use **double angle formula**: $\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$. First, find $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$.<br /><br />4. Apply the double angle formula<br /> Substitute into the formula: $\tan(2\theta) = \frac{2(-\frac{3}{4})}{1-(-\frac{3}{4})^2} = \frac{-\frac{6}{4}}{1-\frac{9}{16}} = \frac{-\frac{3}{2}}{\frac{7}{16}} = -\frac{3}{2} \times \frac{16}{7} = -\frac{24}{7}$.
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