photo taken by Joe Sueyoshi

Trigonometric Functions of Any Angle

Learning objectives

A student will be able to:


In the previous lesson we introduced the six trigonometric functions, and we worked with these functions in two ways: first, in right triangles, and second, for angles of rotation. In this lesson we will extend our work with trig functions of angles of rotation to any angle in the unit circle, including negative angles, and angles greater than 360\;\mathrm{degrees}. In the previous lesson, we worked with the quadrantal angles, and with the angles 30^\circ, 45^\circ, and 60^\circ. In this lesson we will work with angles related to these angles, as well as other angles in the unit circle. One of the key ideas of this lesson is that angles may share the same trig values. This idea will be developed throughout the lesson.

Reference Angles and Angles in the Unit Circle

In the previous lesson, one of the review questions asked you to consider the angle 150^\circ. If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of 30^\circ, across the x-axis.

Notice that 150^\circ makes a 30^\circ angle with the negative x-axis. Therefore we say that 30^\circ is the reference angle for 150^\circ. Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the x-axis. Notice that 30^\circ is the reference angle for many angles. For example, it is the reference angle for 210^\circ and for -30^\circ.

Review Video: Determining trig function values using refence triangles



Review Video: Determining trig function values using the unit circle


Unit Circle: You Try It Animation