Chapter 5.2

Chapter 5.2

5.2 Verifying Trigonometric IdentitiesIn this section you will study techniques for verifying trigonometric identities.

In 5.3 you will study techniques for solving trigonometric identities

*The key to both is your ability to use the fundamental identities and the rules of algebra to rewrite the trigonometric expressions.

Remember:

•Solving an equation is finding the values that make a statement true.

•For example: is true for

This is an example of a conditional equation.

0sin x nx

5.2 Verifying Trigonometric IdentitiesAn equation that is true for all real values in the domain of the

variable is an identity.

For example: is true for all real numbers x. So, it is an identity.

There is no well-defined set of rules to follow in verifying trigonometric identities, but there are some guidelines.

1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. (In other words, look for ways to get the expression into something you recognize)

xx 22 cos1sin

5.2 Verifying Trigonometric Identities3. Look for opportunities to use the fundamental identities.

Remember: Sine and cosine pair up well, as do secants and tangents, and cosecants and cotangents.

4. If step 1 through 3 are not helping try converting all terms to sines and cosines.

5. MAKE AN ATTEMPT! Always try something, it might provide insight for you.

Note: Since you are trying to verify one side is the same as the other you can not cross multiply or add the same quantity to both sides. Do not assume both sides are equal, that is what you are trying to verify!

O.K. Let’s get started!

5.2 Verifying Trigonometric Identities

Verify the identity

22

2

sinsec

1sec

2

2

sec

1sec Start with the left side because it is more complicated

2

2

sec

1)1(tan

2

2

sec

tan

)(costan 22

)(coscos

sin 22

2

2sin

Pythagorean identity

Simplify

Reciprocal identity

Quotient identity

Simplify

*Remember*

There can be more than one way to verify an identity

5.2 Verifying Trigonometric IdentitiesTry #5 pg. 3531

222 sin21sincos Verify the identity

5.2 Verifying Trigonometric IdentitiesCombining Fractions Before Using Identities

Verify the identity

2sec2sin1

1

sin1

1

)sin1)(sin1(

sin1sin1

sin1

1

sin1

1

2sin1

2

2cos

2

2sec2

Add fractions

Simplify


Reciprocal identity


Verify the identity algebraically0

coscos

sinsin

sinsin

coscos

yx

yx

yx

yx

5.2 Verifying Trigonometric IdentitiesVerify the identity xxx 222 tan)1)(cos1(tan Apply the identities before you multiply

)sin)((sec 22 xx

2

cos

sin

x

x

x

x2

2

cos

sin

x2tan

Reciprocal identity

Rule of exponents

Quotient identity


Verify the identity algebraicallyx

x

xxsec

cos

cottan


Verify the identityx

xxx

sin1

costansec

x

x

x

x

x

x

sin1

sin1

sin1

cos

sin1

cos

x

xxx2sin1

sincoscos

x

xxx2cos

sincoscos

x

xx

x

x22 cos

sincos

cos

cos

x

x

x cos

sin

cos

1

xx tansec

Multiply numerator and denominator by (1+sin x)

Multiply


Separate fractions

Simplify

Identities


Verify the identity algebraically

sin

cos1

cos1

sin

5.2 Verifying Trigonometric IdentitiesWork with each side separately

Verify the identity

sin

sin1

csc1

cot2

csc1

1csc

csc1

cot 22

csc1

)1)(csc1(csc

1csc

sin

sin

sin

1

sin

sin1

1csc

Left Side Right Side


Verify the identity algebraically 1tantan1tan

1tan 23

5.2 Verifying Trigonometric IdentitiesEnriched Pre-Calculus

Verify each identity

xxxxa 2224 tansectantan)

xxxxxb sin)cos(coscossin.) 6443

xxx sincos)cos1( 42

xxx sin)cos(cos 64

)1(sectan 22 xx

xxx 222 tansectan

))(tan(tantan 224 xxx

xxxxx sincossincossin 4243

Rewrite as separate factors

Rewrite as separate factors



Multiply

Multiply


Verify the identity

Try #63 pg. 3556

xxxx 3235 tansectantan

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Chapter 5.2