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Chapter 3: Problem 41

Prove that each equation is an identity. \(\frac{\sin 4 t}{4}=\cos ^{3} t \sin t-\sin ^{3} t \cos t\)

Short Answer

Expert verified
Both sides simplify to \(\sin(t) \cos(t)(\cos^2(t) - \sin^2(t)).\)

Step by step solution

01

Recall Trigonometric Identities

Recall the double-angle and multiple-angle identities. Specifically, \(\sin(2x) = 2 \sin(x) \cos(x)\) and \(\cos(2x) = \cos^2(x) - \sin^2(x)\).
02

Express \(\sin(4t)\)

Using the double-angle identity twice, express \(\sin(4t)\): \(\sin(4t) = \sin(2 \cdot 2t) = 2 \sin(2t) \cos(2t) = 2 (2 \sin(t) \cos(t)) (\cos^2(t) - \sin^2(t)) = 4 \sin(t) \cos(t)(\cos^2(t) - \sin^2(t)).\)
03

Simplify LHS

The left-hand side (LHS) of the equation becomes: \(\frac{\sin(4t)}{4} = \sin(t) \cos(t)(\cos^2(t) - \sin^2(t)).\)
04

Express RHS

The right-hand side (RHS) of the equation is \(\cos^3(t) \sin(t) - \sin^3(t) \cos(t).\) Factor out \(\sin(t) \cos(t)\): \(\sin(t) \cos(t)(\cos^2(t) - \sin^2(t)).\)
05

Verify Equality

Both sides simplify to \(\sin(t) \cos(t)(\cos^2(t) - \sin^2(t)).\) Thus, the equation is proved as an identity.

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Key Concepts

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

Double-Angle Identity
The double-angle identities are crucial tools in trigonometry. They simplify expressions involving angles that are twice or half of a given angle. For example, the identity for sine is: \(\sin(2x) = 2 \sin(x) \cos(x)\). Notice how it transforms a single sine function of double angle into a product of sine and cosine of the original angle. Similarly, for cosine, we have: \(\cos(2x) = \cos^2(x) - \sin^2(x)\).
This identity simplifies expressions involving cosine functions of double angles by representing them with squares of sine and cosine. It鈥檚 often utilized in proofs and trigonometric transformations.
Multiple-Angle Identity
Multiple-angle identities generalize double-angle identities to more complex scenarios. They help express trigonometric functions of multiples of an angle in terms of functions of the original angle. You might encounter identities such as: \(\sin(4x) = 2 \sin(2x) \cos(2x) = 4 \sin(x) \cos(x) (\cos^2(x) - \sin^2(x))\).
This follows from applying the double-angle identity twice.
The idea is to break down higher multiples into sums or differences of smaller ones. Understanding these identities aids significantly in solving more complex trigonometric equations and validating identities as you saw in your exercise.
Trigonometric Proof
Proving trigonometric identities involves showing that two sides of an equation are equal by using known identities. For instance, in your exercise, proving \(\frac{\sin(4t)}{4} = \cos^3(t) \sin(t) - \sin^3(t) \cos(t)\) required manipulating both sides using multiple-angle and factorization techniques.
The steps were:
  • Using the double-angle identity for \(\sin(4t)\)
  • Simplifying the left-hand side (LHS)
  • Expressing the right-hand side (RHS)
  • Verifying that both sides matched. Thus, logical application of these identities step-by-step ensures both sides are shown to be equal, confirming the trigonometric identity.

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Most popular questions from this chapter

Use identities to simplify the expression \(\frac{1}{\cos ^{2} x}-\tan ^{2} x\).

For each equation, either prove that it is an identity or prove that it is not an identity. \(\cot \frac{x}{2}-\tan \frac{x}{2}=\frac{\sin 2 x}{\sin ^{2} x}\)

Complete the sum and difference identities for sine. a. \(\sin (x+y)= \)______ b. \(\sin (x-y)=\) ______

Find the exact value of \(\cos (\alpha+\beta)\) if \(\sin \alpha=-7 / 25\) and \(\sin \beta=8 / 17,\) with \(\alpha\) in quadrant IV and \(\beta\) in quadrant II.

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\). \(\cos 2 \alpha=1 / 3\) and \(360^{\circ}<2 \alpha<450^{\circ}\)
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