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Chapter 5: Problem 2

Give exact values for the quantities. Do not use a calculator for any of these exercises-otherwise you will likely get decimal approximations for some solutions rather than exact answers. More importantly, good understanding will come from working these exercises by hand. (a) \(\cos \left(-\frac{3 \pi}{2}\right)\) (b) \(\sin \left(-\frac{3 \pi}{2}\right)\)

Short Answer

Expert verified
(a) \(\cos \left(-\frac{3 \pi}{2}\right) = 0\) (b) \(\sin \left(-\frac{3 \pi}{2}\right) = 1\)

Step by step solution

01

(a) Find the cosine of -3蟺/2

To find the cosine of a negative angle, we can use the property of cosine being an even function: \(\cos(-x) = \cos(x)\) So, we need to find: \(\cos\left(\frac{3 \pi}{2}\right)\) Now, recognize that 3蟺/2 is a quadrantal angle which corresponds to 270掳. At this point, the cosine value is 0, as we are at the point (0, -1) on the unit circle. So, \(\cos \left(-\frac{3 \pi}{2}\right) = 0\).
02

(b) Find the sine of -3蟺/2

To find the sine of a negative angle, we can use the property of sine being an odd function: \(\sin(-x) = -\sin(x)\) So, we need to find: \(-\sin\left(\frac{3 \pi}{2}\right)\) Now, recognize that 3蟺/2 is a quadrantal angle which corresponds to 270掳. At this point, the sine value is -1, as we are at the point (0, -1) on the unit circle. So, \(\sin \left(-\frac{3 \pi}{2}\right) = -(-1) = 1\). The exact values for the given angles are: (a) \(\cos \left(-\frac{3 \pi}{2}\right) = 0\) (b) \(\sin \left(-\frac{3 \pi}{2}\right) = 1\)

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