Problem 43 - Precalculus: A Concise Course

Question

Use the value of the trigonometric function to evaluate the indicated functions. \(\sin t=\frac{1}{2}\) (a) \(\sin (-t)\) (b) \(\csc (-t)\)

Step-by-Step Solution

Answer

The value of \(\sin (-t)\) is \(-\frac{1}{2}\) and the value of \(\csc (-t)\) is \(-2\).

Step 1: Identifying the given function values

First, get the given value of the sine function, which is \(\sin t = \frac{1}{2}\)

Step 2: Evaluate \(\sin (-t)\)

Next, remember that the sine function is an odd function, meaning that \(\sin(-x) = -\sin(x)\). So, given that \(\sin t = \frac{1}{2}\), it follows that \(\sin (-t) = -\frac{1}{2}\)

Step 3: Evaluate \(\csc (-t)\)

Lastly, remember that the cosecant function is the reciprocal of the sine function, so \( \csc(t) = \frac{1}{\sin(t)} \). Since we've already determined that \(\sin(-t) = -\frac{1}{2}\), it follows that \(\csc(-t) = \frac{1}{\sin(-t)} = -2\)