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Chapter 4: Problem 67

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-\frac{3 \pi}{2}$$

Short Answer

Expert verified
The cosine of \(-\frac{3\pi}{2}\) is \(0\), the sine of \(-\frac{3\pi}{2}\) is \(-1\), and the tangent is undefined because it involves division by zero.

Step by step solution

01

Identify Position on Unit Circle

Remember that in the unit circle, an angle of \(0\) or \(2\pi\) radians is at the positive x-axis. A rotation of \(-\frac{3\pi}{2}\) radians indicates that we move the same amount in the opposite direction (clockwise instead of counter-clockwise) of \(\frac{3\pi}{2}\) radians. This position is at the negative y-axis.
02

Find Sine and Cosine

The point where the terminal side of our angle (after rotation) intersects the unit circle gives us the (cos胃, sin胃) coordinates. For \(-\frac{3\pi}{2}\) radians or \(270^\circ\), this point is (0,-1). Thus, \(\cos\left(-\frac{3\pi}{2}\right) = 0\) and \(\sin\left(-\frac{3\pi}{2}\right) = -1\).
03

Find Tangent

Tangent of an angle in the unit circle is given by \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Plugging in our values, we get \(\tan\left(-\frac{3\pi}{2}\right) = \frac{-1}{0}\). But division by zero is undefined, so the tangent of \(-\frac{3\pi}{2}\) is undefined.

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