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Chapter 5: Problem 23

Solve the equation. $$\tan 3 x(\tan x-1)=0$$

Short Answer

Expert verified
\( x = \frac{n\pi}{3}, \frac{\pi}{4} + n\pi \)

Step by step solution

01

Set the factors equal to zero

First, notice that the equation \( \tan 3x (\tan x - 1) = 0 \) can be satisfied if either \( \tan 3x = 0 \) or \( \tan x - 1 = 0 \). So equate these two terms to zero and solve for x separately.
02

Solve for x in \( \tan 3x = 0 \)

In a tan function, the value is zero at \(n\pi\), where n is an integer. Knowing this, we can write the solution as \(3x = n\pi\). To find x, divide by 3. It gives \(x = \frac{n\pi}{3}\).
03

Solve for x in \( \tan x - 1 = 0 \)

To solve for x, first add 1 to both sides to isolate \(\tan x\): \( \tan x = 1 \). The tangent of \( x \) is 1 at \( x = \frac{\pi}{4} + n\pi \) with n being an integer.

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Most popular questions from this chapter

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$\csc ^{2} x+0.5 \cot x-5=0$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sin x+\cos x$$ Trigonometric Equation $$\cos x-\sin x=0$$

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$3 \tan ^{2} x+4 \tan x-4=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

Find the \(x\) -intercepts of the graph. $$y=\sec ^{4}\left(\frac{\pi x}{8}\right)-4$$
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