Ch. 3 - Trigonometric Identities and Equations

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

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 17

Chapter 3, Problem 17

Find all solutions of each equation. tan x = 0

Verified step by step guidance

Recall that the tangent function is defined as \(\tan x = \frac{\sin x}{\cos x}\), and it equals zero when the numerator, \(\sin x\), is zero (provided \(\cos x \neq 0\)).

Set the equation \(\tan x = 0\) equivalent to \(\sin x = 0\) because \(\tan x = 0\) when \(\sin x = 0\).

Find all angles \(x\) where \(\sin x = 0\). The sine function is zero at integer multiples of \(\pi\), so \(x = n\pi\), where \(n\) is any integer.

Write the general solution as \(x = n\pi\), where \(n \in \mathbb{Z}\) (the set of all integers).

If the problem specifies a domain (such as \(0 \leq x < 2\pi\)), list all values of \(x\) within that domain by substituting integer values of \(n\) accordingly.

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.

Definition of the Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In the unit circle, tan x is the ratio of the y-coordinate to the x-coordinate of the point on the circle. Understanding this helps identify where tan x equals zero.

Solutions of tan x = 0 on the Unit Circle

The equation tan x = 0 means the tangent value is zero, which occurs when the sine of x is zero and cosine is nonzero. On the unit circle, this happens at angles where the point lies on the x-axis, specifically at x = nπ, where n is any integer.

General Solution for Trigonometric Equations

Trigonometric equations often have infinitely many solutions due to periodicity. For tan x = 0, the general solution is expressed as x = nπ, where n is any integer, capturing all angles coterminal with the principal solutions.

Recommended video:

4:34

How to Solve Linear Trigonometric Equations

Related Practice

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°

views

Textbook Question

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

Textbook Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 3x/2 + cos x/2

views

Textbook Question

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)

views

Textbook Question

In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°

Textbook Question

Verify each identity. cos² θ (1 + tan² θ) = 1

views