Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 43

Chapter 2, Problem 43

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)

Verified step by step guidance

Recognize that the expression is \( \csc(\tan^{-1}(\frac{\sqrt{3}}{3})) \). The first step is to let \( \theta = \tan^{-1}(\frac{\sqrt{3}}{3}) \), which means \( \tan(\theta) = \frac{\sqrt{3}}{3} \).

Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Assign the opposite side as \( \sqrt{3} \) and the adjacent side as 3 to form a right triangle representing \( \theta \).

Use the Pythagorean theorem to find the hypotenuse \( h \) of the triangle: \( h = \sqrt{(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} \). Simplify \( \sqrt{12} \) to \( 2\sqrt{3} \).

Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), calculate \( \sin(\theta) = \frac{\sqrt{3}}{2\sqrt{3}} \).

Finally, express \( \csc(\theta) \) as the reciprocal of \( \sin(\theta) \), which is \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Simplify this expression to find the exact value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Trigonometric Functions

Inverse trigonometric functions, like tan⁻¹(x), return the angle whose trigonometric ratio equals x. For example, tan⁻¹(√3/3) gives the angle whose tangent is √3/3, which helps in finding the angle measure needed to evaluate other trigonometric functions.

Right Triangle Trigonometry

Right triangle trigonometry relates the sides of a right triangle to its angles using ratios like sine, cosine, and tangent. Knowing the tangent ratio allows you to determine the opposite and adjacent sides, which can then be used to find other ratios such as cosecant.

Reciprocal Trigonometric Functions

Cosecant (csc) is the reciprocal of sine, defined as csc(θ) = 1/sin(θ). Once the angle θ is found, calculating csc(θ) involves finding the sine of θ and then taking its reciprocal to get the exact value.

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Related Practice

Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec(x + π)

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Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)

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Textbook Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin 5π/6)

Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = cos(x + π/2)

Textbook Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)

Textbook Question

In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = cos(x − π/2)