7.1 – Basic Trigonometric Identities and

Equations

5.4.3

Trigonometric Identities

Quotient Identities

tanθ=sinθcosθ cotθ=cosθ

sinθReciprocal Identities

sinθ= 1cscθ

cosθ= 1secθ

tanθ= 1cotθ

Pythagorean Identities

sin2+ cos2 = 1 tan2+ 1 = sec2 cot2+ 1 = csc2

sin2= 1 - cos2

cos2 = 1 - sin2

tan2= sec2- 1 cot2= csc2- 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!

Where did our pythagorean identities come from??

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

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

1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

€

secθ =1

cosθ=

13

4=

43

€

sin2θ + cos2θ =1

sin2θ +35 ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

=1

sin2θ =2525

−9

25

sin2θ =1625

sinθ = ±45

cscθ =1

sinθ=

1± 4

5= ±

54

3.) sinθ = -1/3, find tanθ

4.) secθ = -7/5, find sinθ€

tan2θ +1 = sec2θ

tan2θ +1 = (−3)2

tan2θ = 8

tan2θ = 8

€

tanθ = 2 2

Identities can be used to simplify trigonometric expressions.

Simplifying Trigonometric Expressions

cosθ+sinθ tanθ

=cosθ +sinθ sinθ

cosθ

=cosθ + sin2θ

cosθ

=cos2θ + sin2θ

cosθ= 1

cosθ=secθ

a)

Simplify.

b)cot2θ

1−sin2θ

=cos2θsin2θcos2θ

1

= 1sin2θ

=csc2θ

5.4.5

=cos2θsin2θ× 1

cos2θ

Simplifing Trigonometric Expressions

c) (1 + tan x)2 - 2 sin x sec x

=1+2tanx+tan2x−2sinxcosx

=1+tan2x+2tanx−2tanx=sec2x

d)cscx

tanx+cotx= 1

sinxsinxcosx

+cosxsinx

= 1sinx

sin2x+cos2xsinxcosx

= 1sinx×sinxcosx

1=cosx

= 1sinx1

sinxcosx

=(1+tanx)2 −2sinx 1cosx

Simplify each expression.

€

1sinθ

cossinθ

1sinθ

•sinθcosθ

1cosθ

= secθ€

=cos x1

sin x ⎛ ⎝ ⎜

⎞ ⎠ ⎟ sin xcos x ⎛ ⎝ ⎜

⎞ ⎠ ⎟

=1

€

cos xcos xsin x ⎛ ⎝ ⎜

⎞ ⎠ ⎟+ sin x

cos2 xsin x

+sin2 xsin x

cos2 x + sin2 xsin x

1sin x

= csc x

Simplifying trig Identity

Example1: simplify tanxcosx

tanx cosxsin xcos x

tanxcosx = sin x

Example2: simplifysec xcsc x

sec xcsc x1sin x

1cos x 1

cos xsinx

1= x

=sin xcos x

= tan x

Simplifying trig Identity

Simplifying trig Identity

Example2: simplify cos2x - sin2x

cos xcos2x - sin2x

cos xcos2x - sin2x 1 = sec x

ExampleSimplify:

= cot x (csc2 x - 1)

= cot x (cot2 x)

= cot3 x

Factor out cot x

Use pythagorean identity

Simplify

ExampleSimplify:

Use quotient identitySimplify fraction with LCD

Simplify numerator

= sin x (sin x) + cos xcos x= sin2 x + (cos x)

cos x cos xcos x

= sin2 x + cos2x

cos x = 1

cos x

= sec x

Use pythagorean identity

Use reciprocal identity

Your Turn!Combine fractionSimplify the numeratorUse pythagorean identityUse Reciprocal Identity

Practice

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:

sintancosxxx

=

1seccos

xx

=

1cscsin

xx

=

tan cscSimplify: secx xx

sin 1cos sin

1cos

xx x

x

=

substitute using each identity

simplify

1cos

1cos

x

x

= 1=

Another way to use identities is to write one function in terms of another function. Let’s see an example of this:

2

Write the following expression in terms of only one trig function:

cos sin 1x x This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

2 2sin cos 1x x = 2 2cos 1 sinx x=

2= 1 sin sin 1x x

2= sin sin 2x x

20

(E) Examples• Prove tan(x) cos(x) = sin(x)

RSLSxLS

xxxLS

xxLS

==

=

=

sin

coscossin

costan

21

(E) Examples • Prove tan2(x) = sin2(x) cos-2(x)

LSRSxRSxxRS

xxRS

xxRS

xxRS

xxRS

==

=

=

=

=

=

2

2

2

2

22

22

22

tancossin

cossin

cos1sin

cos1sin

cossin

22

(E) Examples

• Prove tan

tan sin cosx

x x x =

1 1

LS xx

LS xx x

x

LS xx

xx

LSx x x x

x x

LS x xx x

LSx x

LS RS

=

=

=

=

=

=

=

tantan

sincos sin

cossincos

cossin

sin sin cos coscos sin

sin coscos sin

cos sin

1

1

1

2 2

23

(E) Examples

• Prove sin

coscos

2

11x

xx

=

LS xx

LS xx

LS x xx

LS xLS RS

=

=

=

=

=

sincoscoscos

( cos )( cos )( cos )

cos

2

2

1111 1

11