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Chapter 5: Problem 77

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 false. The correct expression according to the identities of trigonometric functions should be \(\sin 2(-u) = -2 \sin u \cos u\). Thus, the sine of double a negative value is equal to the negative of double the product of sine and cosine of the value, not double the positive value.

Step by step solution

01

Understand the provided information

The statement provided states that for a negative number \(u\), \(\sin 2u = -2 \sin u \cos u\) because the sine function is an odd function. The odd function property suggests that for any \(x\), \(\sin(-x) = -\sin(x)\). However, this does not directly apply to the double angle formula.
02

Apply the double angle formula for sine

Apply the known double angle formula for sine, which is \(\sin 2u = 2 \sin u \cos u\). This formula remains valid regardless of the sign of \(u\). For a negative \(u\), this would become \(\sin 2(-u) = 2 \sin(-u) \cos(-u)\). Then, due to the even property of the cosine function, it simplifies to \(\sin 2(-u) = 2 \sin(-u) \cos u\). Finally, applying the sine function's odd property transforms the formula to \(\sin 2(-u) = -2 \sin u \cos u\). However, it is important to note that while the left-hand side changes the argument of \(u\) to \(-u\), the right-hand side does not.
03

Compare with the original statement

The original statement implies that for a negative \(u\), \(\sin 2u = -2 \sin u \cos u\). Comparing this statement with what has been derived, it is clearly incorrect. The correct version should be \(\sin 2(-u) = -2 \sin u \cos u\). So the left-hand side of the statement needs to have the sine function argument as \(-u\) or \(2u\) be negative.

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