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Math  /  Trigonometry

QuestionWrite the product as a sum: 20cos⁡(25r)sin⁡(21r)=20 \cos (25 r) \sin (21 r)=
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Studdy Solution

STEP 1

1. The expression involves trigonometric functions.
2. We can use trigonometric identities to rewrite the product as a sum.
3. The relevant identity is the product-to-sum formula for sine and cosine.

STEP 2

1. Identify the appropriate trigonometric identity.
2. Apply the identity to rewrite the product as a sum.

STEP 3

Identify the product-to-sum identity that applies to the expression cos⁡(A)sin⁡(B) \cos(A) \sin(B) . The identity is:
cos⁡(A)sin⁡(B)=12[sin⁡(A+B)−sin⁡(A−B)] \cos(A) \sin(B) = \frac{1}{2} \left[ \sin(A + B) - \sin(A - B) \right]

STEP 4

Apply the identity to the given expression 20cos⁡(25r)sin⁡(21r) 20 \cos(25r) \sin(21r) .
First, factor out the constant 20:
20⋅cos⁡(25r)sin⁡(21r)=20⋅12[sin⁡((25r+21r))−sin⁡((25r−21r))] 20 \cdot \cos(25r) \sin(21r) = 20 \cdot \frac{1}{2} \left[ \sin((25r + 21r)) - \sin((25r - 21r)) \right]

STEP 5

Simplify the expression inside the brackets:
sin⁡((25r+21r))=sin⁡(46r) \sin((25r + 21r)) = \sin(46r) sin⁡((25r−21r))=sin⁡(4r) \sin((25r - 21r)) = \sin(4r)

STEP 6

Substitute back into the expression:
20⋅12[sin⁡(46r)−sin⁡(4r)] 20 \cdot \frac{1}{2} \left[ \sin(46r) - \sin(4r) \right]
Simplify by multiplying through by 20:
10[sin⁡(46r)−sin⁡(4r)] 10 \left[ \sin(46r) - \sin(4r) \right]
The product as a sum is:
10[sin⁡(46r)−sin⁡(4r)] \boxed{10 \left[ \sin(46r) - \sin(4r) \right]}

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