Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problem 4

Chapter 7, Problem 4

The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.

Verified step by step guidance

Understand the problem: The point \( \left( \frac{\pi}{4}, 1 \right) \) lies on the graph of \( y = \tan x \). This means that when \( x = \frac{\pi}{4} \), \( y = 1 \).

Recall the definition of the inverse tangent function \( y = \tan^{-1} x \): it reverses the roles of \( x \) and \( y \) from the original function \( y = \tan x \).

Since \( y = \tan x \) passes through \( \left( \frac{\pi}{4}, 1 \right) \), the inverse function \( y = \tan^{-1} x \) will pass through the point where the coordinates are swapped, i.e., \( (1, \frac{\pi}{4}) \).

Therefore, the point \( (1, \frac{\pi}{4}) \) lies on the graph of \( y = \tan^{-1} x \).

This step completes the understanding that the inverse function's graph is the reflection of the original function's graph across the line \( y = x \).

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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, such as arctan or tan⁻¹, reverse the effect of their corresponding trigonometric functions. For y = tan⁻¹ x, the output is the angle whose tangent is x, effectively swapping the roles of input and output compared to y = tan x.

Function and Inverse Function Relationship

If a point (a, b) lies on the graph of a function f, then the point (b, a) lies on the graph of its inverse function f⁻¹. This means the coordinates are reversed, reflecting the inverse relationship between the two functions.

Properties of the Tangent Function

The tangent function, tan x, maps angles to real numbers and is periodic with period π. Its inverse, arctan, maps real numbers back to angles within the principal range (-π/2, π/2), ensuring a one-to-one correspondence necessary for defining the inverse.

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Introduction to Tangent Graph

Related Practice

Textbook Question

Find the exact value of each real number y. Do not use a calculator.

y = sin⁻¹ √2/2

Textbook Question

Which one of the following equations has solution π?

a. arccos (―1) = x

b. arccos 1 = x

c. arcsin (―1) = x

Textbook Question

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).

<IMAGE>

sin x = ―√3/2

Textbook Question

Which one of the following equations has solution 3π/4

a. arctan 1 = x

b. arcsin √2/2 = x

c. arccos (―√2 /2) = x

Textbook Question

Decide whether each statement is true or false. If false, explain why.

The tangent and secant functions are undefined for the same values.

Textbook Question

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).

<IMAGE>

sin x = ―1/2