Question

Solve each equation on the interval $\tan (2 \theta)=-1$

   Solve each equation on the interval  
$\tan (2 \theta)=-1$

Algebra and Trigonometry

Algebra and Trigonometry

Michael Sullivan 11th Edition

Chapter 8, Problem 30

Instant Answer

Solved by Expert Raushan Kumar

05/22/2021

Step 1

The tangent function is negative in the second and fourth quadrants. Therefore, we can write $2 \theta = \pi - \frac{\pi}{4}$ and $2 \theta = 2\pi - \frac{\pi}{4}$. Show more…

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Solve each equation on the interval $\tan (2 \theta)=-1$

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Key Concepts

Double-Angle Trigonometric Equations

When the trigonometric function has a multiple angle, such as tan(2?), the equation becomes a double-angle equation. Solving such equations typically involves first solving the equation for the multiple angle and then dividing by the multiplier, while also accounting for the adjustment in the period of the function relative to the variable.

General Approach to Solving Trigonometric Equations

Solving trigonometric equations involves identifying the principal or basic solution(s) from the unit circle and then leveraging the periodicity of the trigonometric function to express the general solution. This approach is applicable across various trigonometric functions and equations.

Periodicity of the Tangent Function

The tangent function repeats its values every ? radians, which means that if tan(?) equals a certain value, then tan(? + k?) will equal the same value for any integer k. This property is essential for determining all possible solutions of trigonometric equations involving tangent.

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