8.4 Relationships Among the Functions Objective 1.To simplify trigonometric expressions and to prove trigonometric identities. 2.To use the fundamental

8.4 Relationships Among the Functions

Objective

1. To simplify trigonometric expressions and to prove trigonometric identities.

2. To use the fundamental identities to find the values of other trigonometric functions from the value of a given trigonometric function.

RECIPROCAL IDENTITIES

sin

1csc

cos

1sec

tan

1cot

QUOTIENT IDENTITIES

cos

sintan

sin

coscot

2 21 tan sec

22 csc cot1

PYTHAGOREAN IDENTITIES

1 cossin 22

Note It will be necessary to recognize alternative forms of the identities above, such as sin² = 1 – cos² and cos² = 1 – sin² .

NEGATIVE IDENTITIES

tan)tan(cos)cos(sin)sin(

csc( ) csc sec( ) sec cot( ) cot

All of the identities we learned are found in the back page of your book under the heading Trigonometric Identities and then Fundamental Identities.

You'll need to have these memorized or be able to derive them for this course.

Pythagorean Identities

Note It will be necessary to recognize alternative forms of the identities above, such as sin² = 1 – cos² and cos² = 1 – sin² .

2 2 2 2 2 2sin cos 1 1 tan sec 1 cot csc

Relationships Among the Functions

2 2 22 2

2 2 2

sin cos sin 11 tan 1 seccos cos cos

2 2 2 2 22 2

2 2 2 2sin cos 1y x x y r

r r r r

2 2 22 2

2 2 2

cos sin cos 11 cot 1 cscsin sin sin

Cofunction Relationships

COFUNCTION IDENTITIES

sin cos cos sin

tan cot cot tan

sec csc csc sec

90 90

90 90

90 90

a f a fa f a fa f a f

and

and

and

COFUNCTION IDENTITIES

sin cos cos sin2 2

tan cot cot tan2 2

sec csc csc sec2 2

and

and

and

Each of the trigonometric relationships given is true for all values of the variable for which each side of the equation is defined.

Such relationships are called trigonometric identities.

1 sin sec sin tan sin

cos cos

xx x x x

x x

21 sin

cos cos

x

x x

21 sin

cos

x

x

2cos

cos

x

x cos x

Example 1: Simplify secx – sinx tanx

[Solution]

Example 2: Write tan + cot in terms of sin and cos .

[Solution]

sincos1

sincoscossin

sincoscos

sincossin

sincos

cossin

cottan

22

22

2

2cot 1 tan

Example 3: Prove: csctan

A AA

A

2cot sec

tan

A A

A21

cot sectan

A AA

21 1sec

tan tanA

A A

22

1sec

tanA

A

22

1cot

cosA

A

2

2 2

cos 1

sin cos

A

A A

2

1

sin A 2csc A

[Proof]

An Identity is NOT a Conditional Equation

• Conditional equations are true only for some values of the variable.

• You learned to solve conditional equations in Algebra by “balancing steps,” such as adding the same thing to both sides, or taking the square root of both sides.

• We are not “solving” identities so we must approach identities differently.

22

4

82

912

2

2

2

xorx

x

x

x

Example 4: Prove

[Proof] Manipulate right to look like left. Expand the binomial and express in terms of sin & cos.

2 2 2(csc cot ) csc 2csc cot cotx x x x x x

x

xxx

cos1

cos1)cot(csc 2

2 2

2 2 2

1 2 cos cos 1 2cos cos

sin sin sin sin sin

x x x x

x x x x x

2 2

2 2

(1 cos ) (1 cos ) (1 cos )(1 cos )

sin 1 cos (1 cos )(1 cos )

x x x x

x x x x

1 cos

1 cos

x

x

1. Learn the fundamental identities.

2. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side.

3. It is often helpful to express all functions in terms of sine and cosine and then simplify the result.

4. Usually, any factoring or indicated algebraic operations should be performed. For example,

5. As you select substitutions, keep in mind the side you are not changing, because it represents your goal.

6. If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x.

. and ,)1(sin1sin2sin cossinsincos

cos1

sin122

We Verify (or Prove) Identities by Doing the Following:

Suggestions• Start with the more complicated side

• Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier)

• Try algebra: factor, multiply, add, simplify, split up fractions

• If you’re really stuck make sure to:Change everything on both sides to sine

and cosine.

• Work with only one side at a time!

Verifying an Identity (Working with One Side)

Example 5: Verify that the following equation is an identity.

cot x + 1 = csc x(cos x + sin x)

Analytic Solution Since the side on the right is more complicated, we work with it.

cot 1 csc (cos sin )1 (cos sin )

sincos sinsin sincot 1

x x x x

x xxx xx xx

Original identity

1cscsin

xx

Distributive property

cos sincot , 1sin sin

x xxx x


[Proof]

tttttt 22 cscsec

cossincottan

tt

tt

tttt

tttt

ttt

ttt

ttt

ttt

tttt

22

22

cscsec

sin1

cos1

cossin1

sincos

cossin1

cossin

cossin1

cotcossin1

tan

cossincot

cossintan

cossincottan

Verifying an Identity (Working with One Side)

Verifying an Identity (Working with Both Sides)


[Proof]

2

2

cos

sinsin21tansectansec

sincos

sincos

sec tan (sec tan )cossec tan (sec tan )cos

sec cos tan cossec cos tan cos1 tan cos1 tan cos1 cos 1 sin1 cos 1 sin

Now work on the right side of the original equation.

We have shown that

sin1sin1

)sin1)(sin1()sin1(

sin1)sin1(

cos)sin1(

cos

sinsin21

2

2

2

2

2

2

2

.cos

sinsin21sin1sin1

tansectansec

2

2

Verifying an Identity (Working with Both Sides)

How to get proficient at verifying identities:

• Once you have proved an identity go back to it, redo the verification or proof without looking at how you did it before, this will make you more comfortable with the steps you should take.

• Redo the examples done in class using the same approach, this will help you build confidence in your instincts!

Don’t Get Discouraged!

• Every identity is different

• Keep trying different approaches

• The more you practice, the easier it will be to figure out efficient techniques

• If a solution eludes you at first, sleep on it! Try again the next day. Don’t give up!

• You will succeed!

3

If the angle is acute (less than 90o) and you have the value of one of the six trigonometry functions, you can find the other five.

Sine is the ratio of which sides of a right triangle?h

o

Draw a right triangle and label and the sides you know.

1

When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem.

a

222 31 a

228 a

22

Now find the other trig functions

cosh

a

22

3sec3

22

Reciprocal of sine so "flip" sine over csc 3

tana

o

22

1

"flipped" cos

cot 22"flipped"

tan

3

1sin

There is another method for finding the other 5 trig functions of an acute angle when you know one function. This method is to use fundamental identities.

3

1sin We'd still get csc by taking reciprocal of

sincsc 3

Now use the trig identity1cossin 22 Sub in the value of sine that you know1cos

3

1 22

Solve this for cos

9

8cos2

3

22

9

8cos

This matches the answer we got with the other method

You can easily find sec by taking reciprocal of cos.

We won't worry about because angle is acute.

square root both sides

3sec

2 2

Let's list what we have so far: 3

1sin

csc 3

We need to get tangent using fundamental identities.

cos

sintan

Simplify by inverting and multiplying

322

31

tan

3

22cos

Finally you can find cot by taking the reciprocal of this answer.

22

3sec

22

3

3

1

22

1

22cot

[Solution]a) sec To find the value of

this function, look for an identity that relates tangent and secant.

Tip: Use Pythagorean Identities.

When is in quadrant II, cos, sec, tan, cot, and csc are all negative.

Example 8: If and is in quadrant II, find each function value.

[Solution]b) sinTip: Use Quotient Identities.

c) cot Tip: Use Reciprocal

Example 8: If and is in quadrant II, find each function value. (Cont.)

1 3cot

tan 5

d) csc Tip: Use Reciprocal

1 34csc

sin 5 34

34

5

[Solution]e) cosTip: Use Reciprocal.

Example 8: If and is in quadrant II, find each function value. (Cont.)

1 3cos

sec 34

3 34

34

Example 9: If and is in quadrant VI, find each function value.

3sin

5

[Solution]a) cosTip: Use Pythagorean

Identities.

22 3

cos 1 sin 15

4

5

When is in quadrant VI, csc, tan, and cot, are all negative, only cos and sec are positive.

b) secTip: Use Reciprocal Identities.

1 5sec

cos 4

c) tanTip: Use Quotient Identities.

sin 3 / 5 3tan

cos 4 / 5 4

Example 9: If and is in quadrant VI, find each function value. (Cont.)

3sin

5

[Solution]d) cotTip: Use Reciprocal Identities.

1 4cot

tan 3

e) cscTip: Use Reciprocal.

1 5csc

sin 3

Example 10: If and find each function value.

1cos

3

Challenge!

[Solution] Since cos > 0, then is either in Quadrant I or VI.

Case 1) If is in Quadrant I , then

a) sin

22 1

sin 1 cos 13

2 2

3

b) tan

sin 2 2 / 3tan 2 2

cos 1/ 3

c) cot

1 1 2cot

tan 42 2

Example 10: If and find each function value. (Cont.)

1cos

3

Challenge!


Case 1) If is in Quadrant I , then

d) sec e) csc1 3 3 2

cscsin 42 2

1sec 3

cos


1cos

3

Challenge!


Case 2) If is in Quadrant VI , then

a) sin

22 1

sin 1 cos 13

2 2

3

b) tan

sin 2 2 / 3tan 2 2

cos 1/ 3

c) cot

1 1 2cot

tan 42 2


1cos

3

Challenge!


Case 2) If is in Quadrant VI , then

d) sec e) csc1 3 3 2

cscsin 42 2

1sec 3

cos

Assignment

P. 321 #1 – 3, 5, 7 – 13, 15 – 23 (odd), 14 – 26 (even), 29 – 33, 35

Documents

8.4 Relationships Among the Functions Objective 1.To simplify trigonometric expressions and to prove trigonometric identities. 2.To use the fundamental