Chapter 4

Analytic Trigonometry

Section 4.1

Trigonometric Identities

Trigonometric Relations

The six trigonometric functions are related in many different ways. Several of these are quite useful for solving different problems, finding values for the trigonometric functions or solving trigonometric equations.

Basic Trigonometric Identities: (They form the building blocks for many other identities.)

x

xx

x

xx

xx

xx

xx

sin

coscot

cos

sintan

tan

1cot

cos

1sec

sin

1csc

xxxxxx 222222 csccot1sec1tan1cossin

Reciprocal Identities:

Pythagorean Identities

Even/Odd Identities

xxxxxx tan)tan(cos)cos(sin)sin( Cofunction Identities

xxxxxx

xxxxxx

seccsctancotsincos

cscseccottancossin

222

222

Simplifying Trigonometric Expressions

It is often useful to be able to simplify a trigonometric expression. For example, when you are trying to solve an equation that involves trigonometric functions. There are many ways to do this but the two that are used very often are:

1. Change to sines and cosines.

2. Combine fractions and expressions where possible.

Here are some examples. (Notice these are expressions! (i.e. there is no equal sign).)

xx csctanSimplify

xx

x

sin

1

cos

sin

xcos

1

xsec

Simplify

xx 22 cot1sin

xx 22 cscsin

xx

22

sin

1sin

1

Simplify

x

xx

cot

sincsc

xx

x x

sincos

sin1 sin

x

xx

xx

x

sin

sinsin

sincos

sin1

x

x

cos

sin1 2

x

x

cos

cos2

xcos

Verifying Identities

A trigonometric identity is different from a trigonometric expression in that an identity will have an equal (=) sign in it. The point here is to justify by showing the algebra steps why the two sides of the equation are equal. There are many ways (techniques) to go about justifying identities and there is not one that is always easiest, but there are a few that are used often.

1. Start with one side of the equation and see if you can change to be the other side of the equation using algebra steps. (Usually you start with the more complicated side of the equation.)

2. Start with an identity you already know is true and do the same thing to both sides of it to get the identity you want to verify.

3. Change things to sines and cosines to see if you can recognize something in the equation.

Most commonly the first techniques is used. It is often the most straight forward in terms of algebra. It is easiest for us to try to take something that is complicated and make it simpler rather than the other way around.

We will do several examples.

Verify: uu

uucot

tan

seccos

Start with the left hand side (LHS).

u

uuLHS

tan

seccos

u

u u

tan

cos cos1

utan

1

ucot

Therefore:

uu

uucot

tan

seccos

Verify:

cscseccottan

Start with the left hand side (LHS). Change to sines and cosines

cottan LHS

sin

cos

cos

sin

cossin

cos

cossin

sin 22

cossin

cossin 22

cossin

1

sin1

cos

1

cscsec

Therefore:

cscseccottan

Verify:

AAA

sin1

1

sin1

1sec2 2

Start with the right hand side (RHS).

AARHS

sin1

1

sin1

1

)sin1)(sin1(

sin1

)sin1)(sin1(

sin1

AA

A

AA

A

)sin1)(sin1(

sin1sin1

AA

AA

A2sin1

2

A2sec2

A2cos

2

Therefore:

AAA

sin1

1

sin1

1sec2 2

Verify:

2tansecsin1

sin1tt

t

t

t

tLHS

sin1

sin1

)sin1)(sin1(

)sin1)(sin1(

tt

tt

t

tt2

2

sin1

sinsin21

t

tt2

2

cos

sinsin21

2)tan(sec ttRHS

tttt 22 tantansec2sec

t

t

t

t

tt 2

2

2 cos

sin

cos

sin

cos

12

cos

1

t

t

t

t

t 2

2

22 cos

sin

cos

sin2

cos

1

t

tt2

2

cos

sinsin21

This one is done a bit differently by simplifying both sides of the equation and showing you get equal expressions.

Therefore, since LHS=RHS 2tansecsin1

sin1tt

t

t

Trigonometric Substitution

There are times (mostly in calculus) when it becomes necessary to replace the value of x in an expression by a trigonometric function to simplify it and make it easier to deal with.

Example:

Substitute 2cos for x in the expression to the right.

(Assume is between 0 and /2)24

1

x

2)cos2(4

1

2cos44

1

)cos1(4

12

2sin4

1

sin2

1

2

csc