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Section 1.3
Reference Angles
Objectives:
1. To find reference angles.
2. To use reference triangles to
determine trigonometric ratios.
Definition
A reference angle is the angle
formed by the terminal ray of
the given angle and the x-axis.
By drawing a perpendicular
segment from any point on the
terminal ray to the x-axis, you
will form a reference triangle.
EXAMPLE 1 Find sin 210º.
y
 = 210º - 180º
= 30º
210°

x
EXAMPLE 1 Find sin 210º.
y
1
sin 210º = 2
- 3
30º
-1
2
x
Practice Question: Find sin 300º
(round to the nearest ten thousandth).
- 3
sin 300º =
2
≈ -.8660
y
1
60º
2
x
- 3
Practice Question: Find
cot 225º.
y
-1
cot 225º =
-1
=1
-1
-1
45º
2
x
EXAMPLE 2 Find tan 61º40.
40
61º40 = (61 + )º = 61.67º
60
tan 61º40 = tan 61.67º ≈ 1.855
Practice Question: Find
cos 81º15 (round to the nearest ten
thousandth).
15
81º15 = (81 + )º = 81.25º
60
cos 81º15 = cos 81.25º ≈ .1521
EXAMPLE 3 Find cos 118º.
cos 118º ≈ -0.4695
Practice Question: Find
sin 207º (round to the nearest ten
thousandth).
sin 207º ≈ -0.4540
EXAMPLE 4 Find csc 63º.
csc 63º = (sin 63º)-1 ≈ 1.1223
TI-83 Calculator steps (use degree mode):
sin(63)-1[ENTER]
Practice Question: Find
sec 126º (round to the nearest ten
thousandth).
sec 126º = (cos 126º)-1 ≈ -1.7013
TI-83 Calculator steps:
cos(126)-1[ENTER]
2nd quad
180° -  = 
or
-=
3rd quad
 - 180° = 
or
-=
1st quad
=
y
x
4th quad
360° -  = 
or
2 -  = 
Homework:
pp. 16-17
►A. Exercises
Give the measure of the reference angle
for each angle.
3. 320°
Since 320° is in the fourth quadrant,
subtract 320° from 360° giving you a
40° reference angle.
►A. Exercises
Give the measure of the reference angle
for each angle.
5
7.
4
Since a 5/4 angle is in the third
quadrant, subtract  from 5/4 giving
you a reference angle of /4.
►A. Exercises
Use a calculator to find the following
ratios.
9. sin 26°20
►A. Exercises
Use a calculator to find the following
ratios.
13. csc 12°18
►A. Exercises
Use a calculator to find the following
ratios.
15. sec 2.75
►B. Exercises
List the angles that have special angles as
reference angles in the given quadrant.
Include quadrantal angles with any
quadrant they bound. Give the positive
measures less than 360°.
17. Third quadrant
►B. Exercises
Use special angles to find the ratios. Do
not use a calculator for these exercises.
19. csc 315°
►B. Exercises
Use special angles to find the ratios. Do
not use a calculator for these exercises.
5
27. cos
3
►B. Exercises
Find the angle measures for 0°    360°
that are associated with the ratios given
here.
29. tan  = 2.081
►B. Exercises
Find the angle measures for 0°    360°
that are associated with the ratios given
here.
31. cot  = 0.0992
►B. Exercises
Find the angle measures for 0°    360°
that are associated with the ratios given
here.
33. sin  = -0.5446
■ Cumulative Review
36. Convert 27° to radians (use ).
■ Cumulative Review
37. Give the radian measure of 27° as
a decimal.
■ Cumulative Review
The legs of a right triangle are 2 units and
7 units in length, and  is the smallest
angle. Find the following trig ratios.
38. tan 
■ Cumulative Review
The legs of a right triangle are 2 units and
7 units in length, and  is the smallest
angle. Find the following trig ratios.
39. sin 
■ Cumulative Review
The legs of a right triangle are 2 units and
7 units in length, and  is the smallest
angle. Find the following trig ratios.
40. cos 