LESSON 31 (4.4 & 4.1) TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
You should learn to:
1. Use reference angles to evaluate trigonometric functions for angles which form 45 -45 -90 or 30 -60 -90
reference triangles.
2. Use a calculator to evaluate trigonometric functions for any angle.
3. Find coterminal angles and their corresponding trig ratios.
Terms to know: standard position (for an angle), initial side, terminal side, positive versus negative angles, reference
angle, reference triangle, coterminal angles.
Section 4.3 dealt with trigonometric functions for acute angles. This section extends the trigonometric functions to any
angle by using reference angles and reference triangles. A discussion of angles (and their measures) in the coordinate
plane is an important prerequisite to finding trig ratios for all possible angles.
An angle is said to be in standard position in the coordinate plane if it is formed by a rotation from the positive xaxis.
Terminal Side (Ray)
initial side The ray where the measurement of an angle starts
(For Trig; usually the x-axis)
Positive Angle (Rotation)
terminal side The ray where the measurement of an angle stops
Initial Side (Ray)
positive angles Couterclockwise rotation
Terminal Side (Ray)
negative angles Clockwise rotation
Negative Angle (Rotation)
The reference angle for an angle in standard position is the acute angle formed by its terminal side and the
horizontal axis (x-axis
Example 1: Sketch angles in standard position having the following measures, and find their reference angles.
b. 150
a. 315
c. 420
60
45
30
A reference angle can be used to form a reference triangle whose trig ratios are the same as those for the actual angle.
A reference triangle is formed by drawing a vertical segment from the terminal side of the angle to the
x-axis, forming a right triangle. For a reference triangle, the hypotenuse is always considered to be positive. A leg is
considered to be positive if it is to the right of or above the origin. It is considered to be negative if it is to the left of or
below the origin.
Example 2: Build reference triangles, and then find the following trig ratios. Be careful with your signs!
2
1
 3
45
1
2
a.
3
1
60
30
1
2
sin 315 
1
2
b. sin(150 ) 
cos 315 
1
2
tan(150 ) 
tan 315 
1
 1
1
1
2
c. cos 420 
1
1

 3
3
sec(150 )  
2
3
1
2
csc 420 
2
3
cot 420 
1
3
Sometimes it is necessary to find trig ratios for angles in standard position which pass through a given point in the
coordinate plane. In such cases, you do not need to find the angle measure in order to find the trig ratios for the angle.
All you need to do is build a reference triangle and use SOH-CAH-TOA. You may need to use the Pythagorean
Theorem to build the reference triangle.
Example 3: Find the three primary trig ratios (sin, cos, and tan) for an angle  in standard position, whose terminal
side passes through the point (2,3) .
a. sin  
3
13
2
b. cos  
13
c. tan   
3
2
x  13
3
(2) 2  32  x 2
4  9  x2
x 2  13
2
x 2  13
x  13
Example 4: Find the values of these trigonometric functions of  , if sin  = 3/5 and cos  < 0.
csc  
5
3
3
4
4
cos  
5
cot  
5
4
3
If an angle does not have a reference angle of 30 , 45 , or 60 , you can still find its trig ratios (in decimal form) by
using a calculator.
Example 5: Use a calculator to find the following:
a. sin(237 )
sin(237 )  .8387
b. cos 612
cos 612  .3090
c. csc112
d.
1
 1.0785
sin112
cot 305
1
 .7002
tan 305
If angles in standard position share the same terminal side, they are called coterminal angles. The angles in the
diagram below have measures of 30 , 390 , and  330 . They are coterminal.
390
30
330
Coterminal angles have measures which differ by multiples of 360 . Thus, you can find angles which are coterminal
to a given angle by adding or subtracting 360 as many times as you wish.
Example 6: Sketch each angle in standard position, and find one negative and one positive coterminal angle
for each.
a. 120
b. 405
120  360
405  360
240 and  480
and there are others
765 and 45 and  315
and there are others
Since coterminal angles share the same terminal side, they form the same reference angles (and reference triangles).
Thus, they have the same trig ratios.
Example 7: Given that sin 31  .5150 , sin 51 =.7771 , and sin 71  .9455 (to 4 decimal place accuracy),
find the following without using a calculator.
a. sin 431
b. sin(329 )
c. sin 771
sin 431  sin(360  71 )
sin(329)  sin(31  360 )
sin 771  sin(720  51 )
sin 71  .9455
sin 31  .5150
sin 51  .7771
ASSIGNMENT 31 Pages 265-268 (Vocabulary Check 3-4, 23, 24, 26, 27, 30, 32, 102, 104 a-c, 113, 114, 118) + Pages
294-295 (Vocabulary Check 1-7, 1a, 4b, 10, 13-16, 18, 38, 48, 53, 54, 65, 66, 72, 77-83, 111) +
Given cos 27  .8910 and cos87  .0523 , find cos(333 ) and cos807 without using a calculator.
Pages 284-286 (14, 48 a,b,d, 68 (degrees only))