Math  /  Trigonometry

QuestionWrite the expression cos4θsinθsin4θcosθ\cos 4 \theta \sin \theta-\sin 4 \theta \cos \theta as a single sine or cosine.

Studdy Solution

STEP 1

1. The expression involves trigonometric identities.
2. The goal is to simplify the expression to a single sine or cosine function.
3. We will use angle sum identities to achieve this.

STEP 2

1. Recognize the expression as a form that can be simplified using angle sum identities.
2. Apply the angle sum identity for sine.

STEP 3

Recognize that the expression cos4θsinθsin4θcosθ\cos 4\theta \sin \theta - \sin 4\theta \cos \theta can be written in the form of a sine angle sum identity:
The sine angle sum identity is:
sin(AB)=sinAcosBcosAsinB \sin(A - B) = \sin A \cos B - \cos A \sin B
In this expression, let A=4θA = 4\theta and B=θB = \theta.

STEP 4

Apply the sine angle sum identity:
cos4θsinθsin4θcosθ=sin(4θθ) \cos 4\theta \sin \theta - \sin 4\theta \cos \theta = \sin(4\theta - \theta)

STEP 5

Simplify the expression inside the sine function:
4θθ=3θ 4\theta - \theta = 3\theta
Thus, the expression simplifies to:
sin(3θ) \sin(3\theta)
The expression cos4θsinθsin4θcosθ\cos 4\theta \sin \theta - \sin 4\theta \cos \theta is equivalent to:
sin(3θ) \sin(3\theta)

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