Find the exact value of the expression. $$ \sin \left[\sin ^{-1} \frac{3}{5}+\cos ^{-1}\left(-\frac{7}{25}\right)\right] $$ $$ \boldsymbol{\operatorname { s i n }}\left[\sin ^{-1} \frac{3}{5}+\cos ^{-1}\left(-\frac{7}{25}\right)\right]=\square $$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Question

UpStudy AI · Accepted Answer

To find the exact value of the expression:

$$
\sin \left[\sin^{-1} \frac{3}{5} + \cos^{-1}\left(-\frac{7}{25}\right)\right],
$$

let's break it down step by step.

1. **Let $$ A = \sin^{-1}\left(\frac{3}{5}\right) $$:**
   - $$\sin A = \frac{3}{5}$$
   - Since $$ A $$ is in the first quadrant, $$\cos A = \sqrt{1 - \sin^2 A} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \frac{4}{5}$$.

2. **Let $$ B = \cos^{-1}\left(-\frac{7}{25}\right) $$:**
   - $$\cos B = -\frac{7}{25}$$
   - Since $$ \cos B $$ is negative, $$ B $$ is in the second quadrant.
   - $$\sin B = \sqrt{1 - \cos^2 B} = \sqrt{1 - \left(-\frac{7}{25}\right)^2} = \frac{24}{25}$$.

3. **Compute $$ \sin(A + B) $$:**
   $$
   \sin(A + B) = \sin A \cos B + \cos A \sin B
   $$
   Substituting the values:
   $$
   = \left(\frac{3}{5}\right) \left(-\frac{7}{25}\right) + \left(\frac{4}{5}\right) \left(\frac{24}{25}\right)
   = -\frac{21}{125} + \frac{96}{125} = \frac{75}{125} = \frac{3}{5}
   $$

Therefore, the exact value of the expression is:

$$
\boxed{\dfrac{3}{5}}
$$
The exact value of the expression is $$ \frac{3}{5} $$.

Find the exact value of the expression.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Trigonometry Questions

Find the exact value of the expression. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)