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Math  /  Trigonometry

QuestionProblem 2 Using sum and difference identities, simplify the following expression. cos⁡(u−v)=cos⁡ucos⁡v+sin⁡usin⁡vcos⁡(q−r)−cos⁡(q+r)cos⁡(u+v)=cos⁡ucov⁡−sin⁡usin⁡v(cos⁡qcos⁡r+sin⁡qsin⁡r)−(cos⁡qcos⁡r−sin⁡qsin⁡r)\begin{array}{lc} \cos (u-v)=\cos u \cos v+\sin u \sin v & \cos (q-r)-\cos (q+r) \\ \cos (u+v)=\cos u \operatorname{cov}-\sin u \sin v & (\cos q \cos r+\sin q \sin r)-(\cos q \cos r-\sin q \sin r) \end{array}

Studdy Solution

STEP 1

1. We are asked to simplify the given trigonometric expression using sum and difference identities.
2. The expression involves cosine sum and difference identities.
3. We will use the identities for cosine of sum and difference: - cos⁡(u−v)=cos⁡ucos⁡v+sin⁡usin⁡v\cos(u - v) = \cos u \cos v + \sin u \sin v - cos⁡(u+v)=cos⁡ucos⁡v−sin⁡usin⁡v\cos(u + v) = \cos u \cos v - \sin u \sin v

STEP 2

1. Identify the relevant trigonometric identities.
2. Apply the identities to simplify the expression.

STEP 3

Identify the relevant trigonometric identities for the expression:
- For cos⁡(q−r)\cos(q - r), use the identity cos⁡(q−r)=cos⁡qcos⁡r+sin⁡qsin⁡r\cos(q - r) = \cos q \cos r + \sin q \sin r. - For cos⁡(q+r)\cos(q + r), use the identity cos⁡(q+r)=cos⁡qcos⁡r−sin⁡qsin⁡r\cos(q + r) = \cos q \cos r - \sin q \sin r.

STEP 4

Apply the identities to the expression cos⁡(q−r)−cos⁡(q+r)\cos(q - r) - \cos(q + r):
(cos⁡qcos⁡r+sin⁡qsin⁡r)−(cos⁡qcos⁡r−sin⁡qsin⁡r)(\cos q \cos r + \sin q \sin r) - (\cos q \cos r - \sin q \sin r)

STEP 5

Simplify the expression by combining like terms:
cos⁡qcos⁡r+sin⁡qsin⁡r−cos⁡qcos⁡r+sin⁡qsin⁡r\cos q \cos r + \sin q \sin r - \cos q \cos r + \sin q \sin r
This simplifies to:
2sin⁡qsin⁡r2 \sin q \sin r
The simplified expression is:
2sin⁡qsin⁡r \boxed{2 \sin q \sin r}

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