Skip to main content
Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblem 7
Chapter 3, Problem 7

Use substitution to determine whether the given x-value is a solution of the equation.
tan2x=−33,x=5π12\(\tan\) 2x = -\(\frac{\sqrt{3}\)}{3}, \(\quad\) x = \(\frac{5\pi}{12}\)

Verified step by step guidance
1
First, substitute the given value of \(x = \frac{5\pi}{12}\) into the left side of the equation \(\tan 2x\). This means calculating \(\tan\left(2 \times \frac{5\pi}{12}\right)\).
Simplify the expression inside the tangent function: \(2 \times \frac{5\pi}{12} = \frac{10\pi}{12} = \frac{5\pi}{6}\), so you need to find \(\tan\left(\frac{5\pi}{6}\right)\).
Recall or use the unit circle to find the exact value of \(\tan\left(\frac{5\pi}{6}\right)\). Remember that \(\frac{5\pi}{6}\) is in the second quadrant where tangent is negative.
Next, evaluate the right side of the equation, which is \(-\frac{\sqrt{3}}{3}\). This is a constant value you can compare with the left side.
Finally, compare the value of \(\tan\left(\frac{5\pi}{6}\right)\) with \(-\frac{\sqrt{3}}{3}\). If they are equal, then \(x = \frac{5\pi}{12}\) is a solution; if not, it is not a solution.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Tangent Function and Its Properties

The tangent function, tan(θ), is defined as the ratio of sine to cosine (sin θ / cos θ). It is periodic with period π, meaning tan(θ + π) = tan(θ). Understanding how to evaluate tangent at specific angles, especially multiples of π, is essential for solving equations involving tan(2x).
Recommended video:
5:43
Introduction to Tangent Graph

Substitution Method in Trigonometric Equations

Substitution involves replacing the variable x with a given value to verify if it satisfies the equation. Here, substituting x = 5π/12 into tan(2x) allows direct evaluation to check if it equals the given expression. This method helps confirm whether the proposed x-value is a solution.
Recommended video:
04:42
Solve Trig Equations Using Identity Substitutions

Simplifying and Evaluating Trigonometric Expressions

Evaluating tan(2x) at x = 5π/12 requires simplifying the angle 2x = 5π/6 and then calculating tan(5π/6). Knowing exact values of tangent at standard angles and simplifying radicals like √3/3 is crucial to compare both sides of the equation accurately.
Recommended video:
6:36
Simplifying Trig Expressions
Related Practice
Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:b. cos 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question
In Exercises 1–60, verify each identity.tan x csc x cos x = 1
2
views
Textbook Question
In Exercises 7–14, use the given information to find the exact value of each of the following:a. sin 2θ15sin θ = -------- , θ lies in quadrant II.17
Textbook Question
In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

tan 2α
4
views
Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

cos3x2sinx2\(\cos\) \(\frac{3x}{2}\) \(\sin\) \(\frac{x}{2}\)

Textbook Question
Use 105° = 135° - 30° to find the exact value of 105°.
4
views