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Chapter 20: Problem 45

Solve the given problems. Express \(\sin 3 x\) in terms of \(\sin x\) only.

Short Answer

Expert verified
\( \sin 3x = 3 \sin x - 4 \sin^3 x \).

Step by step solution

01

Apply the Triple Angle Formula for Sine

We begin by using the triple angle formula for sine, which states that \( \sin 3x = 3 \sin x - 4 \sin^3 x \). This formula will allow us to express \( \sin 3x \) using the sine of \( x \).
02

Simplify the Expression

In the formula \( \sin 3x = 3 \sin x - 4 \sin^3 x \), notice that it is already expressed entirely in terms of \( \sin x \). Therefore, the expression \( \sin 3x \) in terms of \( \sin x \) is \( \sin 3x = 3 \sin x - 4 \sin^3 x \).

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Key Concepts

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

Trigonometric identities
Trigonometric identities are fundamental tools in trigonometry that simplify expressions and solve problems involving angles and lengths. They are equations that relate different trigonometric functions.
These identities are pivotal in manipulating expressions and are frequently used in various applications ranging from geometry to engineering calculations.
  • Key identities include the Pythagorean identity, reciprocal identities, and angle sum and difference identities.
  • The triple angle identity for sine is a prime example, as it expands \( \sin 3x \) into an expression based solely on \( \sin x \).
Understanding and utilizing these identities can greatly simplify complex trigonometric problems.
This simplification allows for a clearer insight into the behavior of trigonometric functions and the relationships between them.
Sine function
The sine function is a cornerstone of trigonometry, representing a periodic wave-like pattern. It is defined based on a right-angled triangle or the unit circle, where the sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a triangle.
In the unit circle, \( \sin \) corresponds to the y-coordinate of a point rotating around the circle's circumference.
  • It has a period of \( 2\pi \) and ranges between \( -1 \) and \( 1 \).
  • The sine function exhibits symmetry, as it is an odd function. This means that \( \sin(-x) = -\sin(x) \).
In the triple angle context, we see the power of the sine function and how it can be expanded using identities to express more complex angles, such as \( \sin 3x \), through simpler representations.
Angle manipulation
Angle manipulation is the process of re-expressing angles, which is crucial in transforming complex expressions into simpler forms. This can involve breaking down composite angles into sums, differences, or multiple angles to uncover underlying relationships within trigonometric functions.
The triple angle formula for sine is a prime example of angle manipulation, transforming \( \sin 3x \) into \( 3\sin x - 4\sin^3 x \).
  • By applying identities like these, we can isolate one trigonometric function to better solve equations and problems.
  • This also aids in graphing, predicting oscillating behaviors, and analyzing periodic phenomena across different domains.
Whether \( 3x \) is broken down from complex curve equations or used in wave functions, manipulating angles provides clearer pathways to solutions, demonstrating the power of mathematical flexibility.

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Most popular questions from this chapter

Prove the given identities. $$\ln (1-\cos 2 x)-\ln (1+\cos 2 x)=2 \ln \tan x$$

Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. $$\tan \left(\frac{3 \pi}{4}+x\right)=\frac{\tan x-1}{\tan x+1}$$

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Solve the given problems. The displacements \(y_{1}\) and \(y_{2}\) of two waves traveling through the same medium are given by \(y_{1}=A \sin 2 \pi(t / T-x / \lambda)\) and \(y_{2}=A \sin 2 \pi(t / T+x / \lambda) .\) Find an expression for the displacement \(y_{1}+y_{2}\) of the combination of the waves.

Prove the given identities. $$1-\cos 2 \theta=\frac{2}{1+\cot ^{2} \theta}$$
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