Download Wiki-Lesson 4.3 - MJNS

Lesson 4.3: Equivalent Trigonometric Expressions
Objectives:
Using Equivalent Trigonometric Expressions to Evaluate Primary /
Reciprocal Trigonometric Expressions
State the co-function formulas (Trigonometric Identities Featuring П/2)
Sin x =
Cos (П/2 – x)
Tan x =
Cot (П/2 – x)
Csc x =
Sec (П/2 – x)
Cos x =
Sin (П/2 – x)
Cot x =
Tan (П/2 – x)
Sec x =
Csc (П/2 – x)
Sin (П/2 + x)
= cos x
Tan (П/2 + x)
= - cot x
Csc (П/2 + x)
= sec x
Cos (П/2 + x)
= -sin x
Cot (П/2 + x)
= - tan x
Sec (П/2 + x)
= - csc x
Example: Given that sin П/5 = 0.5878, use equivalent trigonometric
expressions to evaluate the following, to four decimal places.
a) cos 3П/10
Solution: Since an angle of 3П/10 lies in the first quadrant it can be expressed
as the different between П/2 and an angle a. Let find the measure of angle a.
3П/10 = П/2 – a
Now apply the cofunction identity:
a = П/2 - 3П10
3П/10 = П/2 - a
= 5П/10 - 3П/10
= 2П10 = П/5
cos (3П/10) = cos (П/2 –a)
= sin a
= sin П5
= 0.5878
b) cos 7П/10
Solution:
PRACTICE 4.3:
Name ____________________
Date _____________________
Knowledge
Write each of the following in terms of the cofunction identity:
1. sin

12
2. sin
5
18
6. cos
5. cos
2
5

9
3. sin
5
8
4. sin
5
12
7. cos
7
36
8. cos
2
9
Application
Fill in the blanks with the appropriate function name:
2
  
 _________ 

3
 6 
11
19
 sin
10. _____________
60
60
7
1

11. cos
18 _________ 
9
9. sin
B
For right triangle ABC:
12. If sin A 
of cos B?
3
, what is the value
3


13. If cos A=0.109, what is sin   A  ?
2

11
79
 0.9816 , what is sin
14. If cos
?
180
180
c
a
A
C
b
P’(b,a)
P(a,b)
Thinking
15. The reason for the cofunction relationships can be seen from
the diagram. If the sum of the measures of POA and P ' OA
θ
O

, then P and P’ are symmetric with respect to the line y = x.
2
Also, if P=(a,b), then P’=(b,a) and sin   y-coordinate of P = x-coordinate of P’ =


cos     . Use this information to derive similar cofunction relationships for
2

is
tangent and cotangent, as well as secant and cosecant.
A
3.3.2 HOME ACTIVITY: Equivalent Trigonometric
Expressions (Answers)
Knowledge
Write each of the following in terms of the cofunction identity:
1. sin

5
12
2. sin
5
2
 sin
18
9
6. cos
12
5. cos
 cos
2

 cos
5
10

9
 sin
7
18
3. sin
5

 cos
8
8
4. sin
5

 cos
12
12
7. cos
7
11
 sin
36
36
8. cos
2
5
 sin
9
18
Application
Fill in the blanks with the appropriate function name:
2
  
 cos 

3
 6 
11
19
 sin
10. cos
60
60
7
1

11. cos
18 csc 
9
9. sin
For right triangle ABC:
12.
3
3
13. 0.109
14. 0.9816
Thinking


tan   cot    
2

15.


cot   tan    
2



sec   csc    
2



csc   sec    
2
