Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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Problem 6.4

Textbook Question

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

Identify the given point (\(\frac{\pi}{4}, 1\)) on the graph of \(y = \tan x\).

Understand that the point (\(\frac{\pi}{4}, 1\)) means that when \(x = \frac{\pi}{4}\), \(y = \tan(\frac{\pi}{4}) = 1\).

Recognize that the inverse function \(y = \tan^{-1} x\) swaps the roles of \(x\) and \(y\).

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

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

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Key Concepts

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

Tangent Function

The tangent function, denoted as y = tan x, is a fundamental trigonometric function that relates the angle x in a right triangle to the ratio of the opposite side to the adjacent side. It is periodic with a period of π, meaning it repeats its values every π radians. The function is undefined at odd multiples of π/2, where the cosine value is zero.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as y = tan⁻¹ x, are used to find the angle whose tangent is a given number. The range of the inverse tangent function is limited to (-π/2, π/2), which means it only returns angles in the first and fourth quadrants. This is crucial for determining the corresponding angle for a given tangent value.

Coordinate Transformation

In the context of trigonometric functions, coordinate transformation involves switching between the input-output pairs of a function and its inverse. For the point (π/4, 1) on the graph of y = tan x, the corresponding point on the graph of y = tan⁻¹ x can be found by swapping the x and y coordinates, resulting in the point (1, π/4). This reflects the relationship between a function and its inverse.