trig identities. reciprocal identities or pythagorean identities or quotient identities
TRANSCRIPT
cos
sintan
sin
coscot
Reciprocal Identities
csc
1sin OR
sin
1csc
sec
1cos OR
cos
1sec
cot
1tan OR
tan
1cot
Pythagorean Identities
1cossin 22 OR 22 cos1sin OR 22 sin1cos
22 sectan1 OR 1sectan 22
22 csc1cot OR 1csccot 22
Quotient Identities
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.
Before we do some identities, lets practice substituting and cancelling.
2cos1 1.
Write each expression as a single function or a constant.
Hint: look at trig identities!
2sin
cottan 3. Hint: change to sin and/or cos.
sin
cos
cos
sin1
csctan 5.
sin
1
cos
sin
cos
1 sec
1tancos 7. 2
2seccos
2cos
1cos
cos
1 sec
Handout
Write each expression as a single function or a constant.
2
2
sec
tan1 9.
2
2
2
cos1
cossin
11
cos
cos
sin1
2
2
2
2sin1 2cos
Now we will try some with given ratios.
functions.
tric trigonomefive remaining five theof valuesthe
find II,Quadrant in lies and 13
5cos If 11.
5
13
222 cba 222 135 b
16925 2 b1442 b12b
12
13
12sin
13
5cos
5
12tan
12
13csc
5
13sec
12
5cot
Handout
functions. tric trigonomefive
remaining theof values thefind ,0sin and 3
4sec If 13.
3
4
222 cba 222 43 b
169 2 b72 b7b
7
4
7sin
4
3cos
3
7tan
7
4csc
3
4sec
7
3cot
Handout
7
7
7
4csc
7
74
7
7
7
3cot
7
73
costhenquadrant, third the and 4
5csc If 15.
Handout
4
5csc
5
4sin
3
5
222 cba 222 54 b
2516 2 b92 b3b
4
5
3cos
tan then,0cos and 6.sin If 17.
Handout
10
6sin
8
10
222 cba 222 106 b
10036 2 b642 b8b
6
8
6tan
75.tan
?costan of value theis what angle, acutean is and 4
3sin If 19.
Handout
4
3sin
34
222 cba 222 43 b
169 2 b72 b7b
7
4
7cos
7
3tan
4
7
7
3costan
4
3
cscsin 2.
sin
1sin
1sec 4. 2
2tan
x
x
sec
csc 6.
x
x
cos1
sin1
1
cos
sin
1 x
x
x
x
sin
cos
222 coscotsin 8.
2cot1
2csc
1
xcot
csctancossin 10. 2
sin
1
cos
sincossin 2
2sin
functions.
tric trigonomefive remaining five theof valuesthe
find ,0cos and 25
7sin If 12.
725
222 cba 222 257 b
62549 2 b5762 b24b
24
25
7sin
25
24cos
24
7tan
7
25csc
24
25sec
7
24cot
2
2
sin
tan 14.
1sincossin
2
2
2
22
2
sin
1
cos
sin
2cos
1 2sec