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Chapter 7: Problem 30

Find all solutions of the equation. $$\cos 3 x=\sin 3 x$$

Short Answer

Expert verified
The solutions are \( x = \frac{\pi}{12} + \frac{n\pi}{3} \) for any integer \( n \).

Step by step solution

01

Set the Equation in Tangent Form

The equation \( \cos 3x = \sin 3x \) can be rewritten using the tangent function. If \( \sin 3x = \cos 3x \), then the equivalent is \( \tan 3x = 1 \).
02

Find General Solutions for Tangent

The equation \( \tan \theta = 1 \) occurs where \( \theta = \frac{\pi}{4} + n\pi \) for any integer \( n \). For \( 3x \), it becomes \( 3x = \frac{\pi}{4} + n\pi \).
03

Solve for \(x\)

Solving for \(x\), divide each term in the equation \( 3x = \frac{\pi}{4} + n\pi \) by 3. This gives \( x = \frac{\pi}{12} + \frac{n\pi}{3} \).
04

Provide General Solution

The general solution for the equation is \( x = \frac{\pi}{12} + \frac{n\pi}{3} \). This represents all possible solutions to the original equation, where \( n \) is any integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
In trigonometry, the tangent function is fundamental when solving equations involving angles. The tangent of an angle, often denoted as \( \tan \theta \), is defined as the ratio of the sine and cosine of the angle: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] This relationship is crucial when transforming trigonometric equations. Let's consider the equation \( \cos 3x = \sin 3x \). If you divide both sides by \( \cos 3x \), you get: \[ \frac{\sin 3x}{\cos 3x} = 1 \] This equates to \( \tan 3x = 1 \). Recognizing the tangent function allows us to simplify and solve the equation using trigonometric identities and known values.
General Solutions
Finding the general solutions of trigonometric equations is all about capturing all possible angles that satisfy the given equation. For \( \tan \theta = 1 \), a known solution is \( \theta = \frac{\pi}{4} \), since \( \tan \frac{\pi}{4} = 1 \). However, because the tangent function is periodic with a period of \( \pi \), it repeats every \( \pi \) units. Therefore, all angles \( \theta \) satisfying \( \tan \theta = 1 \) can be expressed as: \[ \theta = \frac{\pi}{4} + n\pi \] where \( n \) is any integer. This format offers a comprehensive set of solutions, ensuring that no solutions are overlooked, given the periodic nature of the tangent function.
Trigonometric Identities
Trigonometric identities are tools used to simplify and verify various trigonometric equations. Understanding these identities is crucial in transitioning a problem into a more solvable form. In our specific case of \( \cos 3x = \sin 3x \), the identity \( \tan 3x = 1 \) arises from the equality of sine and cosine. Some important identities to remember include:
  • Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
  • Reciprocal Identities: \( \sec \theta = \frac{1}{\cos \theta} \), \( \csc \theta = \frac{1}{\sin \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \)
  • Angle Sum and Difference Identities: \( \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \) and similar for cosine and tangent
These identities help in expressing complex equations such as \( \cos 3x = \sin 3x \) in terms of simpler functions, paving the way to find systematic solutions.

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Most popular questions from this chapter

As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction \(F\) of the lunar disc that is lit. When the angle between the sun, earth, and moon is \(\theta\left(0 \leq \theta \leq 360^{\circ}\right),\) then $$F=\frac{1}{2}(1-\cos \theta)$$ Determine the angles \(\theta\) that correspond to the following phases. (a) \(F=0 \quad\) (new moon) (b) \(F=0.25\) (a crescent moon) (c) \(F=0.5\) (first or last quarter) (d) \(F=1 \quad\) (full moon)

If a projectile is fired with velocity \(v_{0}\) at an angle \(\theta,\) then its range , the horizontal distance it travels (in feet), is modeled by the function $$R(\theta)=\frac{v_{0}^{2} \sin 2 \theta}{32}$$ If \(v_{0}=2200 \mathrm{ft} / \mathrm{s},\) what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft away?

Rewrite the expression as an algebraic expression in \(x .\) $$\cos \left(\cos ^{-1} x+\sin ^{-1} x\right)$$

Use a double- or half-angle formula to solve the equation in the interval \([0,2 \pi)\). $$\tan x+\cot x=4 \sin 2 x$$

Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval \([0,2 \pi)\). $$\sin 3 x \cos x-\cos 3 x \sin x=0$$
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