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Chapter 5: Problem 139

Determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \(u, \sin 2 u=-2 \sin u \cos u\).

Short Answer

Expert verified
The statement is true. For a negative number u, sin(2u) indeed equals -2sin(u)cos(u), due to the properties of the sine function being an odd function.

Step by step solution

01

Recapitulate properties of sine function and odd functions

The sine is an odd function meaning sin(-u) = -sin(u). The double angle identity for sine gives: sin(2u) = 2sin(u)cos(u). We will use these properties to analyze the given statement.
02

Apply the properties to the given function

Substitute u with -u in the double angle identity for sine. This gives: sin(2(-u)) = 2sin(-u)cos(-u). Simplify left side using the property of odd function to get: -sin(2u) = 2sin(-u)cos(-u).
03

Check if both sides are equal

Simplify the right side using the property of odd function again: -sin(2u) = -2sin(u)cos(u). As you see, both sides of the equation are equal, which verifies the statement in the exercise.

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