`csc(cos^-1 (sqrt(3)/2))` Find the exact value of the expression. (Hint: Sketch a right triangle.)

Textbook Question

Chapter 4, 4.7 - Problem 63 - Precalculus (3rd Edition, Ron Larson).
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loves2learn | (Level 3) Salutatorian

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First, find the inner expression.

Set `cosx=-sqrt3/2 `  because we need to find what value of x yields `-sqrt3/2 `

Use a unit circle to find this value

=> `x=(5pi)/6 `

Draw a triangle,

Given the problem and using cos rules, we know that short leg is 1, the long leg is `-sqrt3 ` and the hypotenuse is 2.

Therefore, find

=> `csc((5pi)/6)`

Using the triangle,

=> `2/1 `

This simplifies,

=> `2 `

Refer to the link for the triangle.

Triangle

Sources:

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