Section 2: Trigonometry on the Unit Circle

THE UNIT CIRCLE To help us understand the geometric meaning of the trigonometric functions, it is helpful to consider what sinθ and cosθ represent on the unit circle. A unit circle is a circle with radius 1 that is centered at the origin. (i.e. this circle would have the equation x2 + y2 = 1). The 4 quadrants are as labeled below. Angles are measured counter- clockwise starting from the positive x-axis.

CAST RULE The CAST rule is used to help you remember the quadrants in which sin(θ ) cos(θ ) and tan(θ ) are positive.

Quadrant 1 is represented by A therefore all three are positive in that quadrant. Quadrant 2 is represented by S therefore sin(θ ) is positive in that quadrant. Quadrant 3 is represented by T therefore tan(θ ) is positive in that quadrant. Quadrant 4 is represented by C therefore cos(θ ) is positive in that quadrant.

SOHCAHTOA AND SPECIAL ANGLES In trigonometry there are special angles at which you should know the value of the various trigonometric functions. The two special triangles below can be used to help you find these values, but first we need to remember how sine and cosine are defined based on the sides of a right angle triangle. To help us remember we use:

SOH CAH TOA

opposite adjacent opposite sinθ = cosθ = tanθ = hypotenuse hypotenuse adjacent

Now, let’s consider two special triangles:

Using the above triangles and SOHCAHTOA, we end up with the following chart:

θ 0 π π π π

6 4 3 2 sin(θ ) 0 1 1 3 1

2 2 2 cos(θ ) 1 3 1 1 0

2 2 2 tan(θ ) 0 1 1 Undefined 3 3

These values can be used to find the values forcsc(θ ) , sec(θ ) , cot(θ ) and can also be used with the unit circle below (or CAST rule) to find values in other quadrants.

______⎛ 5π ⎞ Example: Evaluate cos⎜ ⎟ ⎝ 6 ⎠

5π Solution: First of all, is in the second quadrant so cosine has a negative value. 6 5π π Next, use the unit circle to determine that is related to the angle . 6 6 ⎛ π ⎞ 3 Finally using the special triangle cos⎜ ⎟ = ⎝ 6 ⎠ 2 ⎛ 5π ⎞ 3 Therefore, cos⎜ ⎟ = − ⎝ 6 ⎠ 2 ______⎛ 5π ⎞ Example: Evaluate csc⎜ ⎟ ⎝ 4 ⎠

5π Solution: First, we know is in the third quadrant so sine has a negative value; 4 consequently, cosecant will also have a negative value. 5π π Next, use the unit circle to determine that is related to the angle . 4 4 ⎛ π ⎞ 1 ⎛ 5π ⎞ 1 Finally using the special triangle sin⎜ ⎟ = so sin⎜ ⎟ = − ⎝ 4 ⎠ 2 ⎝ 4 ⎠ 2 ⎛ 5π ⎞ Therefore, csc⎜ ⎟ = − 2 ⎝ 4 ⎠ ______

For a detailed explanation of some more difficult examples, check out the mini-clips!