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Chapter 5: Problem 33

Write the expression as the sine, cosine, or tangent of an angle. $$\cos 3 x \cos 2 y+\sin 3 x \sin 2 y$$

Short Answer

Expert verified
The expression can be written as \(\cos(5x)\).

Step by step solution

01

Identifying the correct trigonometric identity

In this case, we need to identify the trigonometric identity that fits the expression. Here, we see we have \(\cos(A) \cos(B) + \sin(A) \sin(B)\), which is equivalent to the formula for cosine of the sum of two angles, i.e., \( \cos(A + B)\). In our expression, \(A\) corresponds to \(3x\) and \(B\) corresponds to \(2y\).
02

Applying the trigonometric identity

Apply the identified trigonometric identity and substitute \(3x\) and \(2y\) for \(A\) and \(B\) respectively. This gives us the new expression: \( \cos((3x) + (2y))\).
03

Simplifying the expression

Finally, we combine the terms inside the parenthesis to get a simplified expression. This gives us: \( \cos(5x)\).

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