chapter 5: analytic trigonometry

Chapter 5: Analytic Trigonometry

Section 5.1a: Fundamental IdentitiesHW: p. 451-452 1-7 odd,

27-49 odd

Is this statement true?2 1 11

x xx

This identity is a true sentence, but onlywith the qualification that x must be in thedomain of both expressions.

tan sin cos

If either side of the equality is undefined (i.e., at x = –1), thenthe entire expression is meaningless!!!

The statement is a trigonometric identitybecause it is true for all values of the variable for which bothsides of the equation are defined.

The set of all such values is called the domain of validity ofthe identity.

Basic Trigonometric Identities

1cscsin

Reciprocal Identities1seccos

1cottan

1sincsc

1cossec

1tancot

sintancos

Quotient Identitiescoscotsin

0 is in the domain of validity of exactly three of the basicidentities. Which three?


1cscsin


1cottan

1sincsc

1cossec

1tancot

sintancos


0 For exactly two of the basic identities, one side of the equationis defined at and the other side is not. Which two?


1cscsin


1cottan

1sincsc

1cossec

1tancot

sintancos


0 For exactly three of the basic identities, both sides of theequation are undefined at . Which three?

Pythagorean Identities

cos ,sinP t t

2cos tDivide by :

Recall our unit circle:

(1,0)

PWhat are the coordinates of P?

cost

sint

2 2cos sin 1t t So by the Pythagorean Theorem:

2 2

2 2 2

cos sin 1cos cos cos

t t

t t t 2 21 tan sect t


cos ,sinP t t

2sin tDivide by :

Recall our unit circle:

(1,0)

PWhat are the coordinates of P?

cost

sint

2 2cos sin 1t t So by the Pythagorean Theorem:

2 2

2 2 2

cos sin 1sin sin sin

t t

t t t 2 2cot 1 csct t


tan 5 Given and , find and .

2 2cos sin 1 2 21 tan sec 2 2cot 1 csc

cos 0

We only take the positive answer…why?

sin cos2 2sec 1 tan 21 5 26 sec 26

1cos26

tan 5 sin 5cos

sin 5cos

1sin 526

5sin26

Cofunction Identities

sin cos2

cos sin

2

tan cot2

cot tan

2

sec csc2

csc sec

2

Can you explain why each of these is true???

Odd-Even Identities

sin sinx x cos cosx x

csc cscx x sec secx x

tan tanx x cot cotx x

If , find .cos 0.34

Sine is odd

sin 2

sin sin2 2

Cofunction Identity cos 0.34

Simplifying Trigonometric Expressions

3 2sin sin cosx x xSimplify the given expression.

How can we support this answer graphically???

2 2sin sin cosx x x

sin 1xsin x


2

sec 1 sec 1sin

x xx

Simplify the given expression.

Graphical support?

2

2

sec 1sinxx

2

2

tansin

xx

2

2 2

sin 1cos sin

xx x

2

1cos x

2sec x


1 tan1 cot

xx

Simplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.

sin1coscos1sin

xxxx

cos sincos

sin cossin

x xx

x xx

cos sin sincos sin cosx x xx x x

sincosxx

tan x

Simplifying Trigonometric ExpressionsSimplify the given expressions to either a constant or a basictrigonometric function. Support your result graphically.

2 2

2 2

sec tancos sin

u uv v

11

1


sec tan sec tansec

y y y yy

Use the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

1 sin 1 sincos cos cos cos

1cos

y yy y y y

y

1 sin 1 sin coscos cos 1

y y yy y

Simplifying Trigonometric ExpressionsUse the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

1 sin 1 sin coscos cos 1

y y yy y

21 sin sin sin

cosy y y

y

21 sin

cosyy

2coscos

yy

cos y

Simplifying Trigonometric ExpressionsUse the basic identities to change the given expressions to onesinvolving only sines and cosines. Then simplify to a basictrigonometric function.

2

2 2

sec cscsec csc

x xx x

2

2 2

1 1cos sin1 1

cos sin

x x

x x

2 2

2 2 2

1 cos sincos sin sin cos

x xx x x x

2 2

sinsin cos

xx x

sin1x

sin x

Let’s start with a practice problem…Simplify the expression

cos sin1 sin cos

x xx x

cos cos sin 1 sin1 sin cos cos 1 sin

x x x xx x x x

cos cos sin 1 sin1 sin cos

x x x xx x

2 2cos sin sin1 sin cosx x x

x x

1 sin

1 sin cosx

x x

sec x1cos x

How about somegraphical support?

Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.

1 11 sin 1 sinx x

1 sin 1 sin

1 sin 1 sin 1 sin 1 sinx x

x x x x

1 sin 1 sin1 sin 1 sin

x xx x

2

21 sin x

22sec x

2

2cos x

Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.

1 1sec 1 sec 1x x

sec 1 sec 1sec 1 sec 1x xx x

2

2sec 1x

2

2tan x

22cot x

Combine the fractions and simplify to a multiple of a power of abasic trigonometric function.sin 1 cos1 cos sin

x xx x

22sin 1 cossin 1 cosx xx x

2 2sin cos 2cos 1sin 1 cosx x x

x x

2sin x

2csc x

2 2cos

sin 1 cosx

x x

2 1 cossin 1 cos

xx x

Quick check of your algebra skills!!!

212 8 15x x 8b180a c

10,18

Factor the following expression (without any guessing!!!)

What two numbers have a product of –180 and a sum of 8?

Rewrite middle term: 212 10 18 15x x x Group terms and factor: 2 6 5x x 3 6 5x Divide out common factor: 6 5 2 3x x

Write each expression in factored form as an algebraicexpression of a single trigonometric function.

2sin 3 sin 1x x 2sin 2sin 1x x

e.g.,

sinu xLet

Substitute: 2 2sin 2sin 1 2 1x x u u

Factor: 21u

“Re”substitute for your answer: 2sin 1x


2sin 3 sin 1x x

221 cossec

xx

e.g.,

21 2cos cosx x

21 cos x


2sin 3 sin 1x x

2sin cos 1x x

e.g.,

2sin 1 sin 1x x

sin 1 sin 2x x

2sin sin 2x x sinu xLet 2 2u u 1 2u u


2sin 3 sin 1x x

2 2sec sec tanx x x

e.g.,

2sec 1 sec 1x x

2 2sec sec sec 1x x x 22sec sec 1x x

Write each expression as an algebraic expression of a singletrigonometric function. 2sin 3x

2tan 11 tan

e.g.,

tan 1 tan 11 tan

tan 1

2tansec 1

xx

2sec 1sec 1

xx

sec 1x

sec 1 sec 1sec 1x x

x

chapter 5: analytic trigonometry

Documents

basic identities

pythagorean identities

cofunction identities

given expressions

trigonometric expressionsuse

trigonometric identitybecause

basictrigonometric function

pythagorean theorem