TOPIC 1.2 – ANALYTIC TRIGONOMETRY

1.2.1: The Inverse Sine, Cosine, and Tangent Functions1.2.2: The Inverse Trigonometric Functions1.2.3: Trigonometric Identities1.2.4: Sum and Difference Formulas1.2.5: Double-Angle and Half-Angle Formulas1.2.6: Product-to-Sum and Sum-to-Product Formulas1.2.7: Trigonometric Equations (I)1.2.8: Trigonometric Equations (II)

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2

Review of Properties of Functions and Their Inverses

1.2.1& 1.2.2: The Inverse Sine, Cosine, and Tangent Functions

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1sin

xy sin22

x

y x x y sin sin1

The inverse sine function:The inverse sine function denoted by

sine function from . Thus,

is the inverse of the restricted

                                                                                                                           

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x1sin

x1sin

Finding exact values of

1. Let = xsin2. Rewrite = as x1sin

xsin3. Use the exact values to find the value of that satisfies

2

3sin 1

2

2sin 1

Example: Find the exact value of;1- 2-

5

6

1cos

xy cos x0

The inverse cosine function:The inverse cosine function denoted by restricted cosine function from . Thus,

is the inverse of the

y x x y cos cos1

0 y 1 1xwhere and

Example: Find the exact value of;1)

2

1cos 1

7

8

1tan

xy tan 2 2

y

The inverse tangent function:The inverse tangent function denoted by

restricted tangent function from .Thus,

is the inverse of the

y x x y tan tan1

2 2

y xwhere and

Example: Find the exact value of;1) 1tan 1

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Composition of functions involving inverse trigonometric functions

xx 1sinsin

xx sinsin 1

2,2

Inverse properties1.Sine function:

for every x in the interval [-1,1]

for every x in the interval

xx 1coscos

xx coscos 1 ,0

2. Cosine function:

for every x in the interval [-1,1]

for every x in the interval

xx 1tantan

xx tantan 1

2,2

3. Tangent function:for every real number x

for every x in the interval

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7.0coscos 1 sinsin 1 2coscos 1

Example:1-Find the exact value if possible;a) b) c)

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3tansin 1

2

1sincos 1d) e)

2- If x > 0, write x1tansec as an algebraic expression in x

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1.2.3: Trigonometric Identities

Fundamental trigonometric identities

i ii iii.cscsin

.seccos

.cottan

1 1 1

Reciprocal Identities

i ii. tansin

cos.cot

cos

sin

Quotient Identities

i ii

iii

.sin cos . tan sec

. cot csc

2 2 2 2

2 2

1 1

1

Pythagorean Identities

i ii iii

iv v vi

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

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

Even - Odd Identities

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Example: Verify the identity:Changing to sine and cosine1) xxx sectancsc 2) xxxx cscsincotcos

3) xxxx

xxcossin

cscsec

)csc(sec

Using factoring1) xxxx 32 sincossinsin 2) x

x

x

x

xcsc2

sin

cos1

cos1

sin

Multiplying numerator and denominator by the same factor1)

x

x

x

x

cos

sin1

sin1

cos

Working with both sides separately1)

2tan22sin1

1

sin1

1

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1.2.4: Sum and Difference Formulas

Sum and difference formulas for cosines and sines

cos (A+ B) = cos A cos B - sin A sin Bcos (A - B) = cos A cos B + sin A sin Bsin (A + B) = sin A cos B + cos A sin Bsin (A - B) = sin A cos B - cos A sin B

Example:1.Using difference formula to find the exact value

a) Give the exact value of 6090cos30cos using the sum and difference formula

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5sin

4612

5 b) Find the exact value of using the fact that

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tantan1coscos

cos

5

4sin angle

2

1sin

angle

2. Verify the identity:

3. Suppose that for a quadrant II and

for a quadrant I . Find the exact value ofcos cos

cos sin

a) b)

c) d)

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tan( )tan tan

tan tan

1

tan( )tan tan

tan tan

1

Sum and difference formulas for tangents1-

2-

Example:1.Verify the identity: xx tantan

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1.2.5: Double-Angle and Half-Angle Formulas

cossin22sin

22 sincos2cos

2tan1

tan22tan

Double angle formulas:1- 2-

3-

5

4sin

Example:

1- If and lies in quadrant II, find the exact value of;

2sin 2cos 2tana) b) c)

2. Find the exact value of 15sin15cos 22

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Using Pythagorean identity to write 2cos in terms of sine only:

22 sincos2cos

1cos22cos 2 2sin212cos

Three forms of the double angle formula for cos

1-

2-

3-

Example: Verify the identity: 3sin4sin33sin

2

2cos1sin2

2

2cos1cos2

2cos1

2cos1tan2

Power reducing formulas

x4sin

of trigonometric functions greater than 1

that does not contain powers Example: Write an equivalent expression for

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Half angle formulas

cos1

cos1

2tan;

2

cos1

2cos;

2

cos1

2sin

The + or – in each formula is determined by the quadrant in which 2

lies

Example:1- Use cos 120º to find the exact value of cos 105º2- Verify the identity:

2cos1

2sintan

2

sin

cos1

2tan

cos1

sin

2tan

Half angle formula for tan

Example: Verify the identity:

csccscsec

sec

2tan

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1.2.6: Product-to-Sum and Sum-to-Product Formulas

)]cos()[cos(2

1sinsin

)]cos()[cos(2

1coscos

)]sin()[sin(2

1cossin

)]sin()[sin(2

1sincos

1-

2-

3-

4-

Example: Express each of the following products as a sum or difference:a. sin 5x sin 2x b. cos 7x cos x

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2cos

2sin2sinsin

2cos

2sin2sinsin

2cos

2cos2coscos

2sin

2sin2coscos

Sum to Product Formulas:1-

2-

3-

4-

Example:1- Express each sum or difference as a producta. sin 7x + sin 3x b. cos 3x +cos 2x

2- Verify the identity: xxx

xxtan

sin3sin

cos3cos

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1.2.7& 1.2.8: Trigonometric Equations

• A trigonometric equation is an equation that contains a trigonometric expression. • To solve an equation containing a single trigonometric function:

Isolate the function on one side of the equation Solve for the variable

Finding all solutions of a trigonometric equationExample: Solve the equation: 3sin3sin5 xx

Solving an equation with a multiple angleExample: Solve the equation:

32tan x

20 x

2

1

3sin x

20 x

1-

2-

22

01sin3sin2 2 xx 20 x

xxx sintansin 20 x

Trigonometric equations quadratic in form

Example: Solve the equation:

Using factoring to separate 2 different trigonometric functions in an equation

Example: Solve the equation:

0sin2cos xx 20 x

2

1cossin xx 20 x

1sincos xx 20 x

Using an identity to solve a trigonometric equation

Example: Solve the equation:1-

2-

3-

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