Trig Functions and Graphs

Radians

A radian is a way to measure angles based on how many radiuses lie along the arc intercepted by the central angle. For an angle in radians, where S = arc length:

Since there are pi radians in half a circle, and also 180 degrees in half a circle, you can convert degrees to radians and vice versa with the following proportion:

Also, we are expected to be able to work with revolutions. Recall that 1 revolution = 1 circumference = 360 degrees = 2pi radians.

Since rates are determined by the amount of change divided by time, the following formulas apply:

The Unit Circle

For any point on the unit circle, the x coordinate is the cosine of the central angle, and the y coordinate is the sine of the central angle. The following values should be memorized. Each point (x, y) indicates (cos, sin) of the central angle. (diagram source: The PreCal40S Blog)

Graphs of Trig Functions

Sine and Cosine graphs come in this format: y = a sin (bx) + c where a is the amplitude, b is the frequency (how many complete cycles in 2pi), and c is the vertical shift. The period, or the length of one cycle, can be obtained by .

For example, here is the graph of y = -3 cos (2x) + 4 from -2pi to 2pi, which has a period of pi:

Practice Problems

1. Write in degrees and in number of revolutions.

2. In radians, find the central angle of a 9 cm arc in a 6 cm circle.

3. The second hand on a clock is 6 inches long. Find the speed of the tip of the second hand.

4. Give the period of each trig function: y = sin x, y = cos x, y = tan x.

5. Describe each trig function as odd or even: y = sin x, y = cos x, y = tan x.

6. Find in radians in terms of pi.

7. Find in radical form.

8. Write an equation for a sinusoidal graph with a minima at 0, a maxima at 4, and a period of pi.

9. What is the range of the function y = -5cosx - 3 ?

10. Find the linear speed in cm/sec and angular speed in radians/sec of a point on the edge of a rotating compact disk that makes 500 revolutions per minute. A standard cd is 12 cm in diameter.

(Solutions Pending)