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Chapter 5: Problem 81

Determine whether the statement is true or false. Justify your answer. $$\sin (u \pm v)=\sin u \cos v \pm \cos u \sin v$$

Short Answer

Expert verified
The statement is True. The given function correctly represents the identities for sine of sum and difference of two angles.

Step by step solution

01

Verifying for Addition

Let's first verify the statement for addition, i.e., when \(u + v\). If the statement is true, it should satisfy the identity: \[\sin (u + v) = \sin u \cos v + \cos u \sin v \]which is indeed a well-known trigonometric identity. This identity can be used directly if the values of \(u\) and \(v\) are known. Here, since no specific values are given, a direct verification can be done with the identity itself.
02

Verifying for Subtraction

Now let's verify the statement for subtraction, i.e., when \(u - v\). In this case, the statement should satisfy the identity: \[\sin (u - v) = \sin u \cos v - \cos u \sin v \]Similar to step 1, this is also a known trigonometric identity. This completes the verification of the given statement.
03

Conclusion

Since the given statement satisfies both addition and subtraction scenarios, it can be concluded that the statement is indeed a true representation of the sine of sum or difference of two angles.

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