Transcript Trig Identities - Camden Central School District

Trig Identities
Quotient Identities
sin 
cos 
cos 
cot  
sin 
tan  
Reciprocal Identities
1
sin  
csc 
cos  
1
sec 
tan  
1
cot 
OR
1
csc  
sin 
OR
sec  
1
cos 
OR
cot  
1
tan 
Pythagorean Identities
sin 2   cos2   1 OR
sin 2   1  cos2 
1  tan2   sec2  OR
tan2   sec2   1
cot2   1  csc2  OR
cot2   csc2   1
OR
cos2   1  sin 2 
Tips for proving trigonometric identities:
1. You want to make the left and right hand sides of the identities
match by substitution and cancellation.
2. Work with the more complicated side of the identity.
3. Begin by writing all expressions in terms of sine and/or
cosine.
4. If there is a squared term, check to see if you can use one of
the Pythagorean identities. If so, use it to replace the squared
term.
5. You are finished when the left hand side of the identity
EXACTLY matches the right side. You can not move a term
from one side to the other side.
Handout
Before we do some identities, lets practice substituting and cancelling.
Write each expression as a single function or a constant.
1. 1  cos2 
Hint: look at trig identities!
sin 2 
3. tan cot 
Hint: change to sin and/or cos.
 sin   cos 


 1
 cos  sin  
5. tan csc 
1
 sin   1 


 
 cos  sin   cos 


cos sec  
7. cos tan2   1
 sec 
2
1
 1 
cos 

2
 cos   cos 
 sec 
Handout
Write each expression as a single function or a constant.
sin 2 
2
tan2 
cos


1
9. 1 
1
sec2 
cos2 
sin 2  cos2 
 1

 1 sin 2   cos2 
2
cos 
1
Now we will try some with given ratios.
5
11. If cos   and lies in Quadrant II, find
13
the valuesof thefive remainingfive trigonometric
functions.
12
a 2  b2  c 2
sin


13
52  b 2  132
5
25  b 2  169
cos



13
2
13
b
 144
12
b  12
12
tan




5
5
13
12
13
sec   
5
csc  
cot   
5
12
Handout
4
and sin  0, find the valuesof theremaining
3
five trigonometric functions.
7
sin  
4
3
cos


4
a 2  b2  c 2
32  b 2  42
7
tan


2
9  b  16
3
4
2
b
7
7
b 7

4
7
csc




3
7 7
13. If sec 
cot 
4
7
4
sec  
3
3
cot 
7
csc 
4 7
7
3
7
3 7


7 7
7
Handout
5
15. If csc    and  the third quadrant, then cos  
4
5
4

3
a 2  b2  c 2
4 2  b 2  52
16  b 2  25
b2  9
b3
4
sin   
5
3
cos   
5
5
csc   
4
Handout
17. If sin   .6 and cos  0, thentan 
sin  
10
6

8
a 2  b2  c 2
62  b2  102
36  b 2  100
b2  64
b8
6
10
tan   
6
8
tan   .75
Handout
19. If sin  
3
and  is an acute angle, what is the value of tan  cos  ?
4
3
4
7
cos 
4
3
tan 
7
sin  
4
3

7
a 2  b2  c 2
32  b 2  42
9  b 2  16
b2  7
b 7
3
7
tan cos 

7 4
3

4
Homework
• Handout
#2-20 evens
2. sin  csc 
1
sin  
1
sin 
8. sin 2   cot2   cos2 
1 cot2 
csc2 
4. sec2   1
csc x
sec x
6.
tan2 
1
sin x  1  cos x cos x

1
sin x 1
sin x
cos x
 cot x


10. sin 2  cos tan csc 
sin   1 
 sin  cos  cos


 sin 
2

sin 2 


7
and cos  0, find
25
the valuesof thefive remainingfive trigonometric
functions.
7
a 2  b2  c 2
sin   
2
2
2
25
7  b  25
24
49  b 2  625
cos



25
25
b 2  576
7
b  24
7
tan



24
24
12. If sin   
sin 2 
tan2 cos2 

14.
2
sin 2 
sin 
1
sin 2 
1
1



cos 2 
cos2  sin 2 
 sec2 
25
csc   
7
25
sec   
24
cot  
24
7
16. sin 2   cos2   tan2 
1 tan2 
18. cos  k , then thevalueof cos sin  cot 
 sec2 
cos sin   cos 
 sin  
 cos2   k 2
20. Theexpressionsec2  csc2  is equivalent to
1
1
 2
2
cos  sin 
1  sin 2  
1  cos2 
 2   2 


2 
2 
cos   sin   sin   cos  
sin 2 
cos2

 2
2
2
cos  sin  sin cos2
sin 2   cos2

cos2 sin 2 

1
cos 2 sin 2 