Photo_5.jpgTrigonometric Functions of Any Angle or Real Number

By Tim Brown and Megan Chacosky

DEFINITIONS OF TRIGONOMETRIC FUNCTIONS OF ANY ANGLE:

Use the given point (x,y) as the two sides of the reference triangle. The hypotenuse, r , is the distance from the point (x,y) and the point (0,0). Use the pythagorean theorum to solve for side r. To solve for the value of theta-prime, use the trigonometric functions (sin, cos, tan, csc, sec, cot) to compare two sides with the angle theta-prime. [*The trigonometric functions for angle theta are equal to the trigonometric functions for theta-prime*]
trig_functions_(image_1-given_a_point_(x,y).JPGDon't forget ... SOH-CAH-TOA!

EVALUATING TRIGONOMETRIC FUNCTIONS:

When given a trigonometric function, like tan (theta)=5/4, determine what quadrant to draw the reference triangle in.
trig_functions_(image_2-astc).JPGDon't forget All Students Take Calculus to help remember which functions are positive where!

TRIGONOMETRIC FUNCTIONS OF QUADRANT ANGLES:

trig_functions_(image_3-quadrant_angles).JPGThe four quadrant angles are 0, Π/2, Π, and 3Π/2. Remember that for quadrant angles (x,y) = (cos, sin)

FINDING REFERENCE ANGLES: Let theta be an angle in standard position. Its reference angle is the acute angle theta' by the terminal side of theta and the horizontal axis. Figuring out the reference angle depends on which quadrant theta lies in. In quadrant II theta'=Π-theta(radians) and theta' = 180degrees - theta (degrees). In quadrant III theta' is the opposite as quadrant II. theta' = theta - Π (radians) and theta' = theta - 180 degrees (degrees). In quadrant IV theta' = 2Π-theta (radians) and theta'=360 degrees - theta (degrees).

trig_functions_(image_4-_reference_triangles).JPG

Trigonometric Functions of Real Numbers

The sign of the function value can be determined by the quadrant in which theta lies. To find the value of a trigonometric function of any angle theta: 1. Determine the function value for the associated refereance angle theta. 2. Depending on the quadrant in which theta lies, affix the appropriate sign to the function value.

This table shows trigonometric values of common angles
theta (degrees)
0 degrees
30 degrees
45 degrees
60 degrees
90 degrees
180 degrees
270 degrees
theta (radians)
0
pi/6
pi/4
pi/3
pi/2
pi
3pi/2
sin theta
0
1/2
root 2/2
root 3/2
1
0
-1
cos theta
1
root 3/2
root 2/2
1/2
0
-1
0
tan theta
0
root 3/3
1
root 3
undef.
0
undef.