3.6 Functions of Special and Quadrantal Angles
An angle whose terminal side lies on the x- or y- axis is called a quadrantal angle.
y
(0,1)
(-1,0)
(1,0)
x
(0,-1)
Example 1: Find the values of the six trigonometric functions for each angle.
(a)
𝜋
2
𝜋
sin 2 =
𝜋
cos 2 =
𝜋
(b) 180
(c) 630
𝜋
csc 2 =
𝜋
sec 2 =
𝜋
tan 2 =
cot 2 =
sin 180=
csc 180=
cos 180=
sec 180=
tan 180=
cot 180=
sin 630=
csc 630=
cos 630=
sec 630=
tan 630=
cot 630=
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 1
(d) 0
𝜋
(e) − 2
sin 0=
csc 0=
cos 0=
sec 0=
tan 0=
cot 0=
𝜋
sin (− 2 ) =
𝜋
cos (− 2 ) =
𝜋
(f) 540
𝜋
csc (− 2 ) =
𝜋
sec (− 2 ) =
𝜋
tan (− 2 ) =
cot (− 2 ) =
sin 540=
csc 540=
cos 540=
sec 540=
tan 540=
cot 540=
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 2
Recall from geometry the special triangles:
45
2
1
2
45
60
1
30
1
3
Example 2: Find the exact values of the six trigonometric functions for each.
(a) 45
(b) 30
(c) 60
sin 45=
csc 45=
cos 45=
sec 45=
tan 45=
cot 45=
sin 30=
csc 30=
cos 30=
sec 30=
tan 30=
cot 30=
sin 60=
csc 60=
cos 60=
sec 60=
tan 60=
cot 60=
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 3
Example 3: Find the exact values of the six trigonometric functions for each angle.
(a) 135
(b) 225
𝜋
(c) − 4
sin 135=
csc 135=
cos 135=
sec 135=
tan 135=
cot 135=
sin 225=
csc 225=
cos 225=
sec 225=
tan 225=
cot 225=
𝜋
sin (− 4 ) =
𝜋
cos (− 4 ) =
𝜋
(d) 315
𝜋
csc (− 4 ) =
𝜋
sec (− 4 ) =
𝜋
tan (− 4 ) =
cot (− 4 ) =
sin 315=
csc 315=
cos 315=
sec 315=
tan 315=
cot 315=
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 4
For any angle that is not a quadrantal angle, the reference angle ′, is the acute angle formed by the
terminal side of  and the x-axis.
y
y
y


'
' = 

x

x
x
'

 ' = 180 - 
' =  - 
y
x
'
 ' =  - 180
' =  - 

'
 ' = 360 - 
 ' = 2 - 
To find the reference angle for an angle with negative measure or for an angle greater than 360 (2), first
find a coterminal angle measure whose measure is between 0 and 360 (0 and 2).
Example 4: Give the reference angle ′ for each angle in standard position.
(a) 135
(b) −
7𝜋
3
𝜋
(c) − 4
(d) 110
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 5
Example 5: Use the reference angle to find the sin , cos , and tan  for each value of .
(a) 150
sin 150=
cos 150=
tan 150=
(b) 330
sin 330=
cos 330=
tan 330=
(c)
7𝜋
6
sin
cos
tan
(d) -30
7𝜋
6
7𝜋
6
7𝜋
6
=
=
=
sin (-30)=
cos (-30)=
tan (-30)=
(e)
2𝜋
3
sin
cos
tan
2𝜋
3
2𝜋
3
2𝜋
3
=
=
=
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 6

P(x,y)
'
Q(x',y')
'
The symmetry of a circle centered at the origin provides insight into the relationship between the
trigonometric values of an angle and its reference angle. The angle  and its reference angle ′ intersect
the circle of radius r at P(x, y) and Q(x′, y′). By symmetry x = -x′, and y = y′. Therefore,
𝑥
cos 𝜃 = 𝑟 = −
𝑥′
𝑟
= − cos 𝜃′
sin 𝜃 =
𝑦
𝑟
=
𝑦′
𝑟
𝑦
= sin 𝜃′
𝑦′
tan 𝜃 = 𝑥 = − 𝑥′ = − tan 𝜃′
For any number between -1 and 1, there are infinitely many angles  for which sin  = k. However, it is
often important to determine the values of  for which sin  = k on a restricted domain, such as
0 <  < 2.
Example 6:
1
(a) If 0 <  < 2, determine the values of  for which sin 𝜃 = − 2
1
(b) If 0 <  < 2, determine the values of  for which sin 𝜃 = 2
(c) If 0 <  < 2, determine the values of  for which tan  = 1
Homework: Day 1: p. 156 => Class Exercises 1 – 8; Function Chart
Day 2: p. 157 => Practice Exercises 3 – 18; 20 – 21
Day 3: pp. 157 – 158 => Practice Exercises 24-34; 36; 41-45
Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal Angles
Page 7