Chapter 7: Q 14. (page 457)

In Problems 9–36, find the exact value of each expression.
$\cot [\sin^{- 1} (- \frac{1}{2})]$

Short Answer

Expert verified

The value of the expression $\cot [\sin^{- 1} (- \frac{1}{2})] = - \sqrt{3}$ .

Step by step solution

Step 1. Given information.

The given expression is:

$\cot [\sin^{- 1} (- \frac{1}{2})]$

Let's take $θ = \sin^{- 1} (- \frac{1}{2})$ ,

$\sin θ = - \frac{1}{2}$

$- \frac{π}{2} \leq θ \leq \frac{π}{2}$ is the domain of the function.

Step 2. Find the value of θ.

Initially, we have function $\sin^{- 1}$ and the bounds of sine function are going to be in the interval $[- \frac{π}{2}, \frac{π}{2}]$ . It represents the range of $\sin^{- 1}$ .

By using the unit circle, we can say that must $θ$ be equal to $- \frac{π}{6}$ , so that we can get $- \frac{1}{2}$ .

$\sin θ = - \frac{1}{2} \sin^{- 1} (- \frac{1}{2}) = - \frac{π}{6}$

Step 3. Find the exact value of the expression.

$\cot [\sin^{- 1} (- \frac{1}{2})] = \cot (- \frac{π}{6}) \cot (θ) = \frac{1}{\tan θ} \cot (- \frac{π}{6}) = \frac{1}{\tan (- \frac{π}{6})}$

As we know $\tan θ = \frac{\sin θ}{\cos θ}$ , then

localid="1646422050104" $\tan (- \frac{π}{6}) = \frac{\sin (- \frac{π}{6})}{\cos (- \frac{π}{6})} = - \frac{\sin (\frac{π}{6})}{\cos (\frac{π}{6})} = - \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = - \frac{1}{\sqrt{3}} \tan (- \frac{π}{6}) = - \frac{1}{\sqrt{3}} \cot (- \frac{π}{6}) = \frac{1}{(- \frac{1}{\sqrt{3}})} = - \sqrt{3}$

Therefore, $\cot [\sin^{- 1} (- \frac{1}{2})] = - \sqrt{3}$ .

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In Problems 9–36, find the exact value of each expression.
$\cot [\sin^{- 1} (- \frac{1}{2})]$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the value of θ.

Step 3. Find the exact value of the expression.

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In Problems 9–36, find the exact value of each expression.cotsin-1-12

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the value of θ.

Step 3. Find the exact value of the expression.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Logic and Functions

Decision Maths

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Calculus

Study anywhere. Anytime. Across all devices.

In Problems 9–36, find the exact value of each expression.
$\cot [\sin^{- 1} (- \frac{1}{2})]$