While you wait - · PDF file · 2015-02-06While you wait: For a-d: use a calculator to evaluate: a)sin50 ,cos40 b)sin25 ,𝑐 𝑠65 c)𝑐 𝑠11 ,sin79 d)sin83 ,cos7 Fill in the

While you wait:

For a-d: use a calculator to evaluate:

a) sin 50𝑜, cos 40𝑜

b) sin 25𝑜, 𝑐𝑜𝑠65𝑜

c) 𝑐𝑜𝑠11𝑜, sin 79𝑜

d) sin 83𝑜, cos 7𝑜

Fill in the blank.

a) 𝑠𝑖𝑛30° = cos ___°

b) 𝑐𝑜𝑠57° = sin ___°

Trigonometric Identities and

Equations

Section 8.4

Cofuntion Relationships

x

y

(x,y)

x

y1

UC revisited

Pythagorean Theorem: 𝑥2 + 𝑦2 = 1

x

y

)sin,(cos

sin

cos

1

θ

UC revisited

Pythagorean Theorem:𝑐𝑜𝑠2𝜃 + 𝑠𝑖𝑛2𝜃 = 1

The trig relationships:

𝑐𝑜𝑠2𝜃 + 𝑠𝑖𝑛2𝜃 = 1

𝑐𝑜𝑠2𝜃 = 1 − 𝑠𝑖𝑛2𝜃

𝑠𝑖𝑛2𝜃 = 1 − 𝑐𝑜𝑠2𝜃

• An identity is an equation that is true for all values of the variables.

• Difference between identity and equation:

• An identity is true for any value of the variable, but an equation is not. For example the equation 3x=12 is true only when x=4, so it is an equation, but not an identity.

What are identities used for?

• They are used in simplifying or rearranging algebraic expressions.

• By definition, the two sides of an identity are interchangeable, so we can replace one with the other at any time.

• In this section we will study identities with trig functions.

The trigonometry identities

• There are dozens of identities in the field of trigonometry.

• Many websites list the trig identities. Many websites will also explain why identities are true. i.e. prove the identities.

• For an example of such a site: click here

5.4.3

Trigonometric Identities

Quotient Identities

tan sin

coscot

cos

sin

Reciprocal Identities

sin 1

csccos

1

sectan

1

cot

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

Reciprocal Identity another

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

Reciprocal Identity a third

Pythagorean Identity

Identities can be used to simplify trigonometric expressions.

Simplifying Trigonometric Expressions

cos sin tan

cos sin

sin

cos

cos

sin2

cos

cos 2 sin2

cos

1

cos

sec

a)

Simplify.

b) cot2

1 sin2

cos 2

sin2 cos

2

1

1

sin2

csc2

5.4.5

cos 2

sin2

1

cos2

• Practice Problems for Day 1:

refer to class handout.

While you wait

• Factor:

a) 𝑥2 − 4

b) 𝑥2 − 36

c) 𝑥2 − 1

d) 1 − 𝑥2

• Identify as True or False:

A. cos −𝜃 = −cos (𝜃)

B. sin −𝜃 = −𝑠𝑖𝑛(𝜃)

C. tan −𝜃 = −tan(𝜃)

Proving a Trigonometric Identity:

1. Transform the right side of the identity into the left side,

2. Vice versa (Left side to Right )

We do not want to use properties from algebra

that involve both sides of the identity.

Guidelines for Proving Identities:

1. It is usually best to work on the more complicated side first.

2. Look for trigonometric substitutions involving the basic identities that may help simplify things.

3. Look for algebraic operations, such as adding fractions, the distributive property, or factoring, that may simplify the side you are working with or that will at least lead to an expression that will be easier to simplify.

4. If you cannot think of anything else to do, change everything to sines and cosines and see if that helps.

5. Always keep an eye on the side you are not working with to be sure you are working toward it. There is a certain sense of direction that accompanies a successful proof.

6. Practice, practice, practice!

Prove

𝒄𝒐𝒕𝑨(𝟏 + 𝒕𝒂𝒏𝟐𝑨)

𝒕𝒂𝒏𝑨= 𝒄𝒔𝒄𝟐𝑨

𝐜𝐨𝐭𝐀(𝐬𝐞𝐜𝟐𝐀)

𝐭𝐚𝐧𝐀= 𝒄𝒔𝒄𝟐𝑨 Pythagorean Relationship

𝑐𝑜𝑠𝐴

𝑠𝑖𝑛𝐴(

1

𝑐𝑜𝑠2𝐴)

𝑠𝑖𝑛𝐴

𝑐𝑜𝑠𝐴

=𝑐𝑠𝑐2𝐴

Definition of trig Functions

1𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐴

𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐴

= 𝑐𝑠𝑐2𝐴

Reduce

𝑐𝑜𝑠𝐴

𝑠𝑖𝑛2𝐴𝑐𝑜𝑠𝐴=𝑐𝑠𝑐2𝐴

Def of trig function. 1

𝑠𝑖𝑛2𝐴= 𝑐𝑠𝑐2𝐴

𝑐𝑠𝑐2𝐴 = 𝑐𝑠𝑐2𝐴

Reduce

Practice Problems Day 2

Sec 8- Written Exercises page 321

#13-19 odds; 29-35 odds

Exit Question: #3b the handout.

A complete, step by step solution must be included.

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

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

sec 1

cos1

34

4

3

sin2 cos2 1

sin2 3

5

2

1

sin2 25

259

25

sin2 16

25

sin 4

5

csc 1

sin1

45

5

4

3.) sinθ = -1/3, find tanθ

4.) secθ = -7/5, find sinθ

tan2 1 sec2

tan2 1 (3)2

tan2 8

tan2 8

tan 2 2

Simplifing Trigonometric Expressions

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

1 2 tanx tan2x 2

sinx

cosx

1 tan2x 2tanx 2 tanx

sec2x

d) cscx

tan x cot x

1

sinx

sinx

cos x

cosx

sinx

1

sinx

sin2x cos

2x

sinxcos x

1

sinx

sinx cos x

1

cos x

1

sinx

1

sinx cos x

(1 tanx)2

2 sinx1

cosx

Simplify each expression.

1sin

cossin

1

sinsin

cos

1

cos sec

cos x1

sin x

sin x

cos x

1

cos xcos x

sin x

sin x

cos2 x

sin xsin2 x

sin x

cos2 x sin2 x

sin x

1

sin x csc x

Simplifying trig Identity

Example1: simplify tanxcosx

tanx cosx sin x cos x

tanxcosx = sin x

Example2: simplify sec x csc x

sec x csc x 1

sin x

1 cos x 1

cos x sinx

1 = x

= sin x cos x

= tan x

Simplifying trig Identity

Simplifying trig Identity

Example2: simplify cos2x - sin2x

cos x

cos2x - sin2x

cos x

cos2x - sin2x 1 = sec x

Example Simplify:

= cot x (csc2 x - 1)

= cot x (cot2 x)

= cot3 x

Factor out cot x

Use pythagorean identity

Simplify

Example Simplify:

Use quotient identity

Simplify fraction with

LCD

Simplify numerator

= sin x (sin x) + cos x cos x

= sin2 x + (cos x) cos x

cos x cos x

= sin2 x + cos2x

cos x

= 1 cos x

= sec x

Use pythagorean identity

Use reciprocal identity

Your Turn! Combine

fraction

Simplify the

numerator Use

pythagorean

identity

Use Reciprocal

Identity

Practice

1

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

sin 1

cos sin1

cos

x

x x

x

substitute using each

identity

simplify

1

cos1

cos

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

38

(E) Examples

• Prove tan(x) cos(x) = sin(x)

RSLS

xLS

xx

xLS

xxLS

sin

coscos

sin

costan

39

(E) Examples

• Prove tan2(x) = sin2(x) cos-2(x)

LSRS

xRS

x

xRS

x

xRS

xxRS

xxRS

xxRS

2

2

2

2

2

2

2

2

22

tan

cos

sin

cos

sin

cos

1sin

cos

1sin

cossin

40

(E) Examples

• Prove tan

tan sin cosx

x x x

1 1

LS xx

LSx

x x

x

LSx

x

x

x

LSx x x x

x x

LSx x

x x

LSx x

LS RS

tantan

sin

cos sin

cos

sin

cos

cos

sin

sin sin cos cos

cos sin

sin cos

cos sin

cos sin

1

1

1

2 2

41

(E) Examples

• Prove sin

coscos

2

11

x

xx

LSx

x

LSx

x

LSx x

x

LS x

LS RS

sin

cos

cos

cos

( cos )( cos )

( cos )

cos

2

2

1

1

1

1 1

1

1