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Math 1330 Section 4. Unit Circle Trigonometry
An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise from the initial side. Negative angles are measured clockwise. We will typically use the Greek letter θ to denote an angle.
Angles that have the same terminal side are called coterminal angles. Measures of coterminal angles differ by a multiple of 360° if measured in degrees or by a multiple of 2 π if measured in radians.
If an angle is in standard position and its terminal side lies along the x or y axis, then we call the angle a quadrantal angle.
You will need to be able to work with reference angles. Suppose θ is an angle in standard position and θ is not a quadrantal angle. The reference angle for θ is the acute angle of positive measure that is formed by the terminal side of the angle and the x axis.
We previously defined the six trigonometric functions of an angle as ratios of the lengths of the sides of a right triangle. Now we will look at them using a circle centered at the
origin in the coordinate plane. This circle will have the equation x^2 + y^2 = r^2. If we select a point P ( x , y ) on the circle and draw a ray from the origin through the point, we have created an angle in standard position. The length of the radius will be r.
tan , 0 cot , 0
cos sec
sin csc
y y
x x x
y
x x
x
y y
y
Trig Functions of Quadrantal Angles and Special Angles
You will need to be able to find the trig functions of quadrantal angles and of angles measuring 30 ° , 45 °or 60 °without using a calculator.
Since sin θ = y andcos θ= x , each ordered pair on the unit circle corresponds to
( cos θ ,sin θ)of some angle θ.
We’ll show the values for sine and cosine of the quadrantal angles on this graph:
Using the identities given above, you can find the other four trig functions of an angle, given just sine and cosine. Note that some values are not defined.
You’ll also need to be able to find the six trig functions of the special angles.
For a 30° angle:
( ) cot( 30 ) 3
tan 30
sec 30 2
cos 30
csc 30 2 2
sin 30
For a 60° angle:
Here’s how you can use a reference angle to find the exact value of a trig function for a multiple of one of the special angles.
First, locate the angle in the coordinate plane. Find the reference angle. Determine the signs of the coordinates of the point where the terminal ray of the angle intersects the unit circle.
Next, rewrite the problem in terms of the reference angle.
Finally, use the exact values for the trig functions in the first quadrant or on the positive x and y axes to finish the problem.
Example 1: Sketch each angle in standard position.
a. 210°
b. -135°
c. 3
d. 2
Example 2: Find three angles, two positive and one negative that are coterminal with each angle.
a. 512°
b. 8
Example 3: Name the quadrant in which both conditions are true: cos θ > 0 and cot θ < 0.
Example 4: Let P ( x , y )denote the point where the terminal side of an angle θ intersects
the unit circle. If P is in quadrant I and 13
y = find the six trig functions of angle θ.
Example 7: Sketch an angle measuring 150° in the coordinate plane. Give the coordinates of the point where the terminal side of the angle intersects the unit circle. Then state the six trigonometric functions of the angle.
Example 8: Evaluate each:
a. sin( 240 °)
b. (^)
tan
c. sec( 210 °)
d. (^)
csc
Example 9: Use a calculator to evaluate each of the following to the nearest ten- thousandth.
a. cos( 148 °)
b. (^)
tan
c. (^)
csc
d. sec( − 217 °)
e. sin( 175 °)
f. cot (− 5. 2 )
