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

(-1,0) (1,0) x

(0,-1)

Example 1: Find the values of the six trigonometric functions for each angle.

(b) 180 sin 180= csc 180=

cos 180= sec 180=

tan 180= cot 180=

(c) 630 sin 630= csc 630=

cos 630= sec 630=

tan 630= cot 630= (d) 0 sin 0= csc 0=

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 1 cos 0= sec 0=

tan 0= cot 0=

(f) 540 sin 540= csc 540=

cos 540= sec 540=

tan 540= cot 540=

Recall from geometry the special triangles:

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 2 45 2 1 2 60 1 45 30 1 3

Example 2: Find the exact values of the six trigonometric functions for each.

(a) 45 sin 45= csc 45=

cos 45= sec 45=

tan 45= cot 45=

(b) 30 sin 30= csc 30=

cos 30= sec 30=

tan 30= cot 30=

(c) 60 sin 60= csc 60=

cos 60= sec 60=

tan 60= cot 60=

Example 3: Find the exact values of the six trigonometric functions for each angle.

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 3 (a) 135 sin 135= csc 135=

cos 135= sec 135=

tan 135= cot 135=

(b) 225 sin 225= csc 225=

cos 225= sec 225=

tan 225= cot 225=

(d) 315 sin 315= csc 315=

cos 315= sec 315=

tan 315= cot 315=

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.

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 4 y y y y

    x ' x x x ' '  '

' ' ' ' = 180 - = - 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

(d) 110

Example 5: Use the reference angle to find the sin , cos , and tan  for each value of .

(a) 150 sin 150=

cos 150=

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 5 tan 150=

(b) 330 sin 330=

cos 330=

tan 330=

(d) -30 sin (-30)=

cos (-30)=

tan (-30)=

 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,

Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 6

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: (a) If 0 <  < 2, determine the values of  for which

(b) If 0 <  < 2, determine the values of  for which

(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 AnglesPage 7