Use the appropriate reciprocal identity to find the function value: \csc\theta, given that...

Question:

Use the appropriate reciprocal identity to find the function value: {eq}\quad \csc\theta, \; {/eq} given that {eq}\displaystyle\;\sin\theta = -\frac{8}{43} {/eq}.

Trigonometric Identities:

Some of the important trigonometric identities are as mentioned below:

{eq}\begin{align} \displaystyle \sin \theta = \frac{1}{\csc \theta}\\ \end{align} {/eq}

{eq}\begin{align} \displaystyle \cos \theta = \frac{1}{\sec \theta}\\ \end{align} {/eq}

{eq}\begin{align} \displaystyle \tan \theta = \frac{1}{\cot \theta}\\ \end{align} {/eq}

Answer and Explanation:

Using the first trigonometric identities as mentioned below:

{eq}\begin{align} \displaystyle \sin \theta = \frac{1}{\csc \theta}\\ \end{align} {/eq}

Hence {eq}\begin{align} \displaystyle \csc \theta = \frac{1}{\sin \theta}\\ \end{align} {/eq}

Here according to the question,

{eq}\begin{align} \displaystyle \sin \theta = -\frac{8}{43}\\ \end{align} {/eq}

Hence {eq}\begin{align} \displaystyle \csc \theta = -\frac{43}{8}\\ \end{align} {/eq}

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Trigonometric Identities: Definition & Uses

from Honors Precalculus Textbook

Chapter 23 / Lesson 1
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