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Chapter 5: Problem 78

Determine whether the statement is true or false. Justify your answer. \(\sin \frac{u}{2}=-\sqrt{\frac{1-\cos u}{2}}\) when \(u\) is in the second quadrant.

Short Answer

Expert verified
The statement is False. The \(\sin \frac{u}{2}\) of a number in the first quadrant (where \(\frac{u}{2}\) lies given \(u\) is in the second quadrant) should be positive.

Step by step solution

01

Statement Evaluation

The given statement is \(\sin \frac{u}{2} = - \sqrt{\frac{1 - \cos u}{2}}\). According to the trigonometric property of half-angles, we know that \(\sin \frac{u}{2} = 卤 \sqrt{\frac{1 - \cos u}{2}}\). The sign depends on the quadrant that \(\frac{u}{2}\) lies in.
02

Determine the Quadrant for \(\frac{u}{2}\)

We are given that \(u\) is in the second quadrant, \(90掳 < u < 180掳\). Therefore, \(\frac{u}{2}\) will lie in the first quadrant, \(45掳 < \frac{u}{2} < 90掳\). In this quadrant, all trigonometric functions are positive.
03

Comparing the Signs

The given statement has a negative sign on the right hand side, while according to the trigonometric property of half-angles, the \(\sin \frac{u}{2}\) in the first quadrant will be positive.
04

Final Conclusion

Since the signs on both sides of the equation do not match, we can conclude that the given statement is false.

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