Trig Identities

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

Homework

• Handout

#2-20 evens

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

222 tancossin 16.

2tan1 2sec

cotsincos of value then the,cos 18. k

sin

cossincos 2cos 2k

toequivalent is cscsec expression The 20. 22

22 sin

1

cos

1

2

2

22

2

2 cos

cos

sin

1

sin

sin

cos

1

22

2

22

2

cossin

cos

sincos

sin

22

22

sincos

cossin

22 sincos

1