For the following exercises, find the reference angle, the...

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.
$$
\frac{5 \pi}{6}
$$

Key Concepts

Cosine Function

The cosine function of an angle is defined as the x-coordinate of the corresponding point on the unit circle. Like sine, cosine is a periodic function with values between -1 and 1, and it directly corresponds to the horizontal distance from the origin to the point on the circle.

Calculator Approximation (Rounding)

In many trigonometric computations, especially when dealing with angles that do not correspond to standard positions on the unit circle, it is often necessary to use a calculator to approximate the values of sine, cosine, or other trigonometric functions. Rounding these values to a specified number of decimal places is a common practice to facilitate clarity and precision in the final answer.

Sine Function

The sine function of an angle in trigonometry is defined as the y-coordinate of the corresponding point on the unit circle. It is a periodic function that oscillates between -1 and 1 and its value is determined by the vertical position of the angle’s terminal side.

Quadrants in the Coordinate Plane

The coordinate plane is divided into four quadrants, each determined by the signs of the x and y coordinates. The quadrant in which the terminal side of an angle lies determines the sign (positive or negative) of the trigonometric functions. Recognizing in which quadrant an angle terminates helps assign the correct sign to the computed values of sine and cosine.

Unit Circle

The unit circle is a circle of radius one centered at the origin of the coordinate plane. It provides a geometric interpretation of trigonometric functions where each point on the circle corresponds to an angle's terminal side; the x-coordinate represents the cosine value, and the y-coordinate represents the sine value.

Reference Angle

The reference angle is the acute angle formed between the terminal side of an angle (in standard position) and the horizontal axis. It is used to simplify the computation of trigonometric functions because the values for sine and cosine for the reference angle are the same in magnitude as for the original angle, differing only in sign based on the quadrant.

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