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Chapter 5: Problem 13

Evaluate \(\sin \left(-\sin ^{-1} \frac{3}{13}\right)\).

Short Answer

Expert verified
The short answer is: \(\sin\left(-\sin^{-1} \frac{3}{13}\right) = -\frac{3}{13}\).

Step by step solution

01

Use arcsine function to find the angle

In this step, we will use the arcsine function to find the angle whose sine value is equal to the given value. So, let \(\theta = \arcsin\left(\frac{3}{13}\right)\). This means that \(\sin(\theta)=\frac{3}{13}\).
02

Find the negative of the angle

Now that we have found the angle \(\theta\), we will find the negative of that angle, which is denoted by \(-\theta\). This can be easily achieved by negating the angle found in step 1. So, \(-\theta = -\arcsin\left(\frac{3}{13}\right)\).
03

Use sine function on the negative angle

Now that the negative angle is found, we will use the sine function to evaluate the original expression given in the exercise. Therefore, the result is: \(\sin\left(-\sin^{-1} \frac{3}{13}\right) = \sin(-\theta) = \sin\left(-\arcsin\left(\frac{3}{13}\right)\right)\). Since \(\sin(\theta)=\frac{3}{13}\), we can use the property \(\sin(-\theta) = -\sin(\theta)\) to find the result, \(\sin\left(-\arcsin\left(\frac{3}{13}\right)\right) = -\sin(\theta) = -\frac{3}{13}\).

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