Find the exact value of the trigonometric function at the...

Find the exact value of the trigonometric function at the given real number.
(a) $\sin \left(-\frac{3 \pi}{2}\right) \quad$ (b) $\cos \left(-\frac{3 \pi}{2}\right) \quad$ (c) $\cot \left(-\frac{3 \pi}{2}\right)$

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, representing the y-coordinate of a point on the unit circle corresponding to a given angle. It has a period of 2?, and it is classified as an odd function, meaning that sin(-?) = -sin(?). These properties are crucial when evaluating sine for any angle, including negative angles.

Cosine Function

The cosine function is another primary trigonometric function, representing the x-coordinate of a point on the unit circle corresponding to a given angle. With a period of 2?, it is considered an even function, which implies that cos(-?) = cos(?). This symmetry is important for calculating cosine values for negative angles.

Cotangent Function

The cotangent function is defined as the ratio of the cosine function to the sine function, i.e., cot(?) = cos(?)/sin(?), provided sin(?) is not zero. It inherits properties from both sine and cosine, including periodicity and behavior under angle negation. Understanding its definition and relation to sine and cosine is key to evaluating cotangent at any angle.

Even-Odd Properties

The even-odd properties refer to how trigonometric functions respond when their input angles are negated. Specifically, sine (and tangent) are odd functions, so they change sign with the negation of the angle, while cosine is even, staying the same. These properties simplify the evaluation of trigonometric functions at negative angles.

Unit Circle

The unit circle is a fundamental tool in trigonometry that visually and conceptually relates angles to coordinates. Each angle corresponds to a point on the circle where the x-coordinate gives the cosine value and the y-coordinate gives the sine value. This concept is essential for understanding and computing the values of trigonometric functions for any angle.

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