Trigonometric Identities

Simplifying

Some Vocab

1. Identity: a statement of equality between two expressions that is true for all values of the variable(s)

2. Trigonometric Identity: an identity involving trig expressions.

Reciprocal Identities

Quotient Identities

Know these!

Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:

Simplify:

tan xcsc x

sec x

substitute using each identity

sintan

cos

xx

x

1sec

cosx

x

1csc

sinx

x

Simplify:

tan xcsc x

sec x

=

sinxcosx

⋅ 1sinx

1cosx

substitute using each identity

Now, Simplify:

One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:

sintan

cos

xx

x

1sec

cosx

x

1csc

sinx

x

tan cscSimplify:

sec

x x

x

=

sinxcosx

⋅ 1sinx

1cosx

substitute using each identity

simplify

1cos

1cos

x

x

1

Do you remember the Unit Circle?

• What is the equation for the unit circle?x2 + y2 = 1

• What does x = ? What does y = ? (in terms of trig

functions)sin2θ + cos2θ = 1

Pythagorean Identity!

cos2θ =1−sin2θ or sin2θ =1−cos2θALSO USED AS:

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by cos2θ

sin2θ + cos2θ = 1 .cos2θ cos2θ cos2θ tan2θ + 1 = sec2θ

Quotient Identity

ReciprocalIdentityanother

Pythagorean Identity

Take the Pythagorean Identity and discover a new one!

Hint: Try dividing everything by sin2θ

sin2θ + cos2θ = 1 .sin2θ sin2θ sin2θ 1 + cot2θ = csc2θ

Quotient Identity

ReciprocalIdentitya third

Pythagorean Identity

IF ONE WANTS A TAN, HE SEEKS THE SUN

IF ONE WANTS TO LOOK LIKE COTTON, HE COVERS WITH SUN SCREEN

QUOTIENT IDENTITIESsin

tancos

xx

x

coscot

sin

xx

x

1+ tan2 x =sec2 x2 21 cot cscx x

PYTHAGOREAN IDENTITIES2 2sin cos 1x x

RECIPROCAL IDENTITIES

1csc

sinx

x

1sec

cosx

x

1cot

tanx

x

1sin

cscx

x

1cos

secx

x

1tan

cotx

x

Opposite Angle Identitiessometimes these are called even/odd identities

Simplify each expression.

Simplify each expression.

1

sinθcosθsinθ

1sinθ

•sinθcosθ

=1

cosθ

=secθ

Simplify each expression.

Simplify the expressio .

cosθ1

•1

sinθ•

sinθcosθ

=1

cosθ cscθ tanθRemember sin and cos:

Simplify with addition.

cosθ cotθ +sinθcosθ1

•cosθsinθ

+sinθ1

Here we need a Common denominator!Simplify here

Now: Prove the identity:

cosθ cotθ +sinθ = cscθcosθ1

•cosθsinθ

+sinθ1

Work on the complicated side only

Now: Prove the identity:

cosθ cotθ +sinθ =cscθcosθ1

•cosθsinθ

+sinθ1

cos2θsinθ

+sinθ1

sinθsinθ

⎛

⎝⎜

⎞

⎠⎟

=cos2θ +sin2θ

sinθ

=1

sinθ=cscθ

Common denominator

Pythagorean identity

Work on the complicated side only

Substituting and factoring:

• Ex: 1−2sinx+(1−cos2 x)

Substituting and factoring:

• Ex: 1−2sinx+(1−cos2 x)

1−2sinx+sin2 x

sin2 x−2sinx+1

Think: x2 −2x+1

Substituting and factoring:

• Ex: 1−2sinx+(1−cos2 x)

1−2sinx+sin2 x

sin2 x−2sinx+1

(sin x −1)(sinx−1)

Substitute, distribute, simplify

• Ex:

cos x −2sin2 x+1

Factor

• Ex: cos x −2sin2 x+1

cosx−2(1−cos2 x)+1

cosx−2+2cos2 x+1

2cos2 x+cosx−1

Factor

• Ex: cos x −2sin2 x+1

cosx−2(1−cos2 x)+1

cosx−2+2cos2 x+1

2cos2 x+cosx−1(2cosx−1)(cosx+1)

Factoring to simplify:

• Ex: 1−sin2 x1+sinx

Factoring to simplify:

• Ex: 1−sin2 x1+sinx(1+sinx)(1−sinx)

1+sinx

1−sinx

Do Now:

• Calculate: cos 60

Cos (A – B)

• From the reference sheet:• COS (A-B)=cosAcosB + sinAsinB• Example:

cos(90−30)=cos90cos30 +sin90sin30

COS (A – B)

• From the reference sheet:• COS (A-B)=cosAcosB + sinAsinB• Example:

• = + = 0 + ½ = ½

cos(90−30)=cos90cos30 +sin90sin300( )

32

⎛

⎝⎜

⎞

⎠⎟ 1( )

12

⎛

⎝⎜

⎞

⎠⎟

Example

• Find cos 15 exactly by using cos(45-30)

Example

• Find cos 15 exactly by using cos(45-30)

cos 45cos 30+sin45sin302

2

⎛

⎝⎜

⎞

⎠⎟

32

⎛

⎝⎜

⎞

⎠⎟+

22

⎛

⎝⎜

⎞

⎠⎟12

⎛

⎝⎜

⎞

⎠⎟

6

4+

24

=6 + 24

Do Now:

• If

• Hint: draw a triangle.

sin A =513

, find the cosA

Do Now:

• IfIf A and B are 2nd quadrant angles, find cos(A-B)

sin A =513

, and cosB=−45

Do Now:

• IfIf A and B are 2nd quadrant angles, find cos(A-B)

sin A =513

, and cosB=−45

Sum and Difference

• Use sin (a+b)• To find the exact value of 75 degrees.

sin(45+ 30)=sin45cos30 +cos45sin30

Sum and Difference

• Use sin (a+b)• To find the exact value of 75 degrees.

sin(45+ 30)=sin45cos30 +cos45sin30

2

2

⎛

⎝⎜

⎞

⎠⎟

32

⎛

⎝⎜

⎞

⎠⎟+

22

⎛

⎝⎜

⎞

⎠⎟12

⎛

⎝⎜

⎞

⎠⎟

=64

⎛

⎝⎜

⎞

⎠⎟+

24

⎛

⎝⎜

⎞

⎠⎟

=6 + 24

Remember the do now?

• Use the same strategy for the next problem.

Finding the missing function first

• Given that

• Write out what you know so far…..• sinA cosB – cosA sinB

Finding the missing function first

• Given that

sinA cosB – cosA sinB

Find the missing sides of both triangles A

B5

13

4

5

Finding the missing function first

• Given that

sinA cosB – cosA sinB5

13

⎛

⎝⎜

⎞

⎠⎟45

⎛

⎝⎜

⎞

⎠⎟−

1213

⎛

⎝⎜

⎞

⎠⎟35

⎛

⎝⎜

⎞

⎠⎟

2065

−3665

⎛

⎝⎜

⎞

⎠⎟=

−1665

AB

513

4

5

12

3cos A =1213

sin B =35

Using the identities you now know, find the trig value.

If cosθ = 3/4, If cosθ = 3/5, find secθ. find cscθ.

sinθ = -1/3, 180o < θ < 270o; find tanθ

secθ = -7/5, π < θ < 3π/2; find sinθ

Do Now:

• Simplify:

1− 13

1+ 13

⎛

⎝⎜

⎞

⎠⎟

Do Now:

• Simplify:

• Now Rationalize:

1− 13

1+ 13

⎛

⎝⎜

⎞

⎠⎟

=3−13 +1

33

33

Do Now:

• Rationalize:

3 −13 +1

•3 −13 −1

=3−2 3 +1

3−1

=4 −2 3

2=2− 3

Tan(A±B)

• Use the reference sheet:• Find tan(45-30)

Tan(A±B)

• Use the reference sheet:• Find tan(45-30)

tan A−tanB1+ tanAtanB

tan45−tan301+ tan45 tan30

Tan(A±B)

• Use the reference sheet:• Find tan(45-30)

tan A−tanB1+ tanAtanB

tan45−tan301+ tan45 tan30

=1− 1

3

1+ 1( )13

⎛

⎝⎜

⎞

⎠⎟

=3−13 +1

Tan(A±B)

• Use the reference sheet:• Find tan(180+45)

tan A+ tanB1−tanAtanB

tan180 + tan451−tan180 tan45

Tan(A±B)

• Use the reference sheet:• Find tan(180+45)

tan A+ tanB1−tanAtanB

tan180 + tan451−tan180 tan45

= 0+11−0(1)

=1

examples

• Answer # 1 on page 502.

sin2A

• Find sin2A if A =45

• 2sinAcosA=

sin2A

• Find sin2A if A =45

• 2sinAcosA=

• And the sin 90 = 1!

22

2

⎛

⎝⎜

⎞

⎠⎟

22

⎛

⎝⎜

⎞

⎠⎟

=44=1

Cos A

• Find cos 2A if A = 30

cos2A =cos2 A−sin2 A

Cos A

• Find cos 2A if A = 30

cos2A =cos2 A−sin2 A3

2

⎛

⎝⎜

⎞

⎠⎟

2

−12

⎛

⎝⎜

⎞

⎠⎟2

=34−14=12

Which quadrant?

• If

• Remember to use a triangle to find the cos.

3 5

Do Now: hint: Which quadrant?

• If

Do Now: hint: Which quadrant?

• If

sin2θ =2−35

⎛

⎝⎜

⎞

⎠⎟45

⎛

⎝⎜

⎞

⎠⎟

=−2425

sin ½ A, cos ½ A

• See the reference sheet:

• Find sin ½ A if A = 60

sin 12θ =±

1−cosθ2

cos12θ =±1+cosθ

2

sin ½ A, cos ½ A

• Find sin ½ A if A = 60 (must be positive – in quadrant I.

sin 12 A =

1− 12

⎛⎝⎜

⎞⎠⎟

2

=

122=

14=12

cos ½ A

• Find cos ½ A if A = 60 (must be positive – in quadrant I.

cos 12θ =

1+cosθ2

cos ½ A

• Find cos ½ A if A = 60 (must be positive – in quadrant I.

cos 12 A =

1+ 12

⎛⎝⎜

⎞⎠⎟

2

=

322=

34=

32

tan ½ A

• Find tan ½ A if A = 90 (must be positive – in quadrant I.

tan 12 A =

1−cosA1+cosA

tan ½ A

• Find tan ½ A if A = 90 (must be positive – in quadrant I.

tan 12 A =

1−01+0

=11=1

Which quadrant?

• If (quadrant III)

• Then (cos is negative in quad. III)

• find

sinθ =−35

and 180 <θ < 270

90< ½θ <135 which is in quadrant II where sin is positive

cosθ =−45

sin 12θ

3 5

4

Which quadrant?

• If

=1−

−4

52

=

9

52

=9

10=

3

10

Or.. 3 10

10

Blow pop question

• Find sin(A+B) if

• Hint: draw two triangles.

sin A =513

, cosB=35

Blow pop question

• Find sin(A+B) if sin A =513

, cosB=35

cos A =1213

, sinB=45