BASIC IDENTITIES

Examples which are worked in this lesson (Click on the number):
Examples Use Basic Identities to find the exact value of the other five
trigonometric functions for the following.
1. csc 5 and is in the III quadrant
2.
3
2
cos and csc 0
3.
8
17
tan and sin 0
Examples Find all the exact solutions for the following equations.
1. 2 cos 1 sin 2
2.
2 7 2cos 8 sin
3. sec 9 7 tan 2
4. 5cot 6csc 5 2 x x
Recall the following Basic Identities from Lesson 2:
1.
cos
sin
tan 2.
sin
cos
cot
3.
tan
1
cot 4.
cos
1
sec
5.
sin
1
csc
Recall the following Pythagorean Identities from Lesson 2:
1. cos sin 1 2 2
2. sec tan 1 2 2
3. csc cot 1 2 2
Examples Use Basic Identities to find the exact value of the other five
trigonometric functions for the following.
1. csc 5 and is in the III quadrant Back to Examples List
Since csc 5 , then
5
1
sin
Since csc 5 and csc cot 1 2 2
by one of the Pythagorean
Identities, then
( 5) cot 1 2 2
25 cot 1 2
cot 24 2
cot 24 . Since is in the III quadrant, then cot 24 .
Since cot 24 , then
24
1
tan .
Now, we need to find cos and sec . Since
5
1
sin and
cos sin 1 2 2
by one of the Pythagorean Identities, then you could
use this identity to find cos . However, you only need to use the
Pythagorean Identities once to solve these problems.
Since
sin
cos
cot , then cos cot sin . Since cot 24 and
5
1
sin , then
5
24
5
1
cos 24 .
Since
5
24
cos , then
24
5
sec .
Answers:
5
24
cos ,
5
1
sin ,
24
1
tan ,
24
5
sec ,
and cot 24
2.
3
2
cos and csc 0 Back to Examples List
First, determine what quadrant the angle is in. Using Method 1 from
Lesson 6, we have:
cos 0 the x-coordinate of the point of intersection of the terminal
side of the angle with the Unit Circle is positive. That is, x > 0.
csc 0 s in 0 the y-coordinate of the point of intersection
of the terminal side of the angle with the Unit Circle is negative. That is,
y < 0.
Thus, we have that x > 0 and y < 0. Thus, the angle is in the IV quadrant.
You may use Method 2 or Method 3 from Lesson 6 if you wish.
Since
3
2
cos , then
2
3
sec
Since
3
2
cos and cos sin 1 2 2
by one of the Pythagorean
Identities, then
sin 1
3
2 2
2
sin 1
9
4 2
9
5
sin 2
3
5
sin . Since is in the IV quadrant, then
3
5
sin .
Since
3
5
sin , then
5
3
csc .
Now,