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 32

Chapter 2, Problem 32

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ sin⁻¹ (− √3/2)

Verified step by step guidance

Recognize that the expression is asking for the angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) and \( \theta \) is in the range of the inverse sine function, which is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

Recall the reference angle where \( \sin(\theta) = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \) because \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).

Since the sine value is negative, and the inverse sine function outputs angles only between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), the angle must be in the fourth quadrant (negative angle).

Therefore, the angle \( \theta \) is the negative of the reference angle: \( \theta = -\frac{\pi}{3} \).

Conclude that \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \) as the exact value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given value. Its output range is limited to angles between -π/2 and π/2 (or -90° to 90°) to ensure it is a function. Understanding this range is crucial for finding the correct angle without a calculator.

Exact Values of Sine for Special Angles

Certain angles have well-known sine values expressed in terms of square roots, such as sin(π/3) = √3/2. Recognizing these exact values allows you to identify the angle corresponding to a given sine value without approximation or a calculator.

Sign and Quadrant Considerations for Inverse Trigonometric Functions

Since sin⁻¹ returns angles only in the first and fourth quadrants, the sign of the sine value determines the angle's sign. For negative sine values, the angle lies between -π/2 and 0. This helps in correctly assigning the angle's sign and value when evaluating inverse sine expressions.

Recommended video:

4:28

Introduction to Inverse Trig Functions

Related Practice

Textbook Question

In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

views

Textbook Question

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

Textbook Question

In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 1/2 csc x/2

Textbook Question

In Exercises 29–36, find the length x to the nearest whole unit.

Textbook Question

In Exercises 29–36, find the length x to the nearest whole unit.

views

Textbook Question

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos⁻¹ √5/7