Analytic Trigonometry

Chapter 6

The Inverse Sine, Cosine, and Tangent Functions

Section 6.1

One-to-One Functions

A one-to-one function is a function f such that any two different inputs give two different outputs

Satisfies the horizontal line test

Functions may be made one-to-one by restricting the domain

Inverse Functions

Inverse Function: Function f {1 which undoes the operation of a one-to-one function f.

Inverse Functions For every x in the domain of f,

f {1(f(x)) = x and for every x in the domain of f {1,

f(f {1(x)) = x Domain of f = range of f {1, and

range of f = domain of f {1

Graphs of f and f {1, are symmetric with respect to the line y = x

If y = f(x) has an inverse, it can be found by solving x = f(y) for y. Solution is y = f {1(x)

More information in Section 4.2

2

32

2

32

-4

-2

2

4

Inverse Sine Function

The sine function is not one-to-one

We restrict to domain

Inverse Sine Function

Inverse sine function: Inverse of the domain-restricted sine function

-4 -2 2 4

2

32

2

32

Inverse Sine Function

y = sin{1x means x = sin yMust have {1 · x · 1 andMany books write y = arcsin x WARNING!

The {1 is not an exponent, but an indication of an inverse function

Domain is {1 · x · 1Range is

Exact Values of the Inverse Sine Function

Example. Find the exact

values of:

(a) Problem:

Answer:

(b) Problem:

Answer:

Approximate Values of the Inverse Sine

Function Example. Find approximate

values of the following. Express the answer in radians rounded to two decimal places.

(a) Problem:

Answer:

(b) Problem:

Answer:

Inverse Cosine Function

Cosine is also not one-to-oneWe restrict to domain [0, ¼]

2

32

2

32

-4

-2

2

4

Inverse Cosine Function

Inverse cosine function: Inverse of the domain-restricted cosine function

-4 -2 2 4

2

32

2

32

Inverse Cosine Function

y = cos{1x means x = cos yMust have {1 · x · 1 and 0 · y

· ¼ Can also write y = arccos x Domain is {1 · x · 1Range is 0 · y · ¼

Exact Values of the Inverse Cosine Function

Example. Find the exact values

of:

(a) Problem:

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

Approximate Values of the Inverse Cosine

Function Example. Find approximate

values of the following. Express the answer in radians rounded to two decimal places.

(a) Problem:

Answer:

(b) Problem:

Answer:

Inverse Tangent Function

Tangent is not one-to-one (Surprise!)

We restrict to domain

2

32

22

32

2

-6

-4

-2

2

4

6

Inverse Tangent Function

Inverse tangent function: Inverse of the domain-restricted tangent function

-4 -2 2 4

2

32

2

32

Inverse Tangent Function

y = tan{1x means x = tan yHave {1 · x · 1 and Also write y = arctan x Domain is all real numbersRange is

Exact Values of the Inverse Tangent

FunctionExample. Find the exact

values of:

(a) Problem:

Answer:

(b) Problem:

Answer:

The Inverse Trigonometric Functions [Continued]

Section 6.2

Exact Values Involving Inverse Trigonometric

FunctionsExample. Find the exact values

of the following expressions

(a) Problem:

Answer:

(b) Problem:

Answer:

Exact Values Involving Inverse Trigonometric

FunctionsExample. Find the exact values

of the following expressions

(c) Problem:

Answer:

(d) Problem:

Answer:

Inverse Secant, Cosecant and

Cotangent Functions Inverse Secant Function

y = sec{1x means x = sec y j x j ¸ 1, 0 · y · ¼,

2

32

22

32

2

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

2

32

2

2

32

2

Inverse Secant, Cosecant and

Cotangent Functions Inverse Cosecant Function

y = csc{1x means x = csc y j x j ¸ 1, y 0

2

32

22

32

2

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

2

32

2

2

32

2

Inverse Secant, Cosecant and

Cotangent Functions Inverse Cotangent Function

y = cot{1x means x = cot y{1 < x < 1, 0 < y < ¼

2

32

22

32

2

-6

-4

-2

2

4

6

-6 -4 -2 2 4 6

2

32

2

2

32

2

Inverse Secant, Cosecant and

Cotangent FunctionsExample. Find the exact values

of the following expressions

(a) Problem:

Answer:

(b) Problem:

Answer:

Approximate Values of Inverse Trigonometric

Functions Example. Find approximate

values of the following. Express the answer in radians rounded to two decimal places.

(a) Problem:

Answer:

(b) Problem:

Answer:

Key Points

Exact Values Involving Inverse Trigonometric Functions

Inverse Secant, Cosecant and Cotangent Functions

Approximate Values of Inverse Trigonometric Functions

Trigonometric Identities

Section 6.3

Identities

Two functions f and g are identically equal provided f(x) = g(x) for all x for which both functions are defined

The equation above f(x) = g(x) is called an identity

Conditional equation: An equation which is not an identity

Fundamental Trigonometric IdentitiesQuotient Identities

Reciprocal Identities

Pythagorean Identities

Even-Odd Identities

Simplifying Using Identities

Example. Simplify the following expressions.

(a) Problem: cot µ ¢ tan µ

Answer:

(b) Problem:

Answer:

Establishing Identities

Example. Establish the

following identities

(a) Problem:

(b) Problem:

Guidelines for Establishing Identities

Usually start with side containing more complicated expression

Rewrite sum or difference of quotients in terms of a single quotient (common denominator)

Think about rewriting one side in terms of sines and cosines

Keep your goal in mind – manipulate one side to look like the other

Key Points

IdentitiesFundamental Trigonometric

IdentitiesSimplifying Using IdentitiesEstablishing IdentitiesGuidelines for Establishing

Identities

Sum and Difference Formulas

Section 6.4

Sum and Difference Formulas for Cosines

Theorem. [Sum and Difference Formulas for Cosines]

cos(® + ¯) = cos ® cos ¯ { sin ® sin ¯

cos(® { ¯) = cos ® cos ¯ + sin ® sin ¯

Sum and Difference Formulas for Cosines

Example. Find the exact values

(a) Problem: cos(105±)

Answer:

(b) Problem:

Answer:

Identities Using Sum and Difference

Formulas

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

Sum and Difference Formulas for Sines

Theorem. [Sum and Difference Formulas for Sines]

sin(® + ¯) = sin ® cos ¯ + cos ® sin ¯

sin(® { ¯) = sin ® cos ¯ { cos ® sin ¯

Sum and Difference Formulas for Sines

Example. Find the exact values

(a) Problem:

Answer:

(b) Problem: sin 20± cos 80± { cos

20± sin 80±

Answer:

Sum and Difference Formulas for Sines

Example. If it is known that

and that

find the

exact values of: (a) Problem: cos(µ + Á)

Answer:(b) Problem: sin(µ { Á)

Answer:

Sum and Difference Formulas for Tangents

Theorem. [Sum and Difference Formulas for Tangents]

Sum and Difference Formulas With Inverse

FunctionsExample. Find the exact

value of each expression(a) Problem:

Answer:

(b) Problem:

Answer:

Sum and Difference Formulas With Inverse

FunctionsExample. Write the

trigonometric expression as an algebraic expression containing u and v.Problem: Answer:

Key Points

Sum and Difference Formulas for Cosines

Identities Using Sum and Difference Formulas

Sum and Difference Formulas for Sines

Sum and Difference Formulas for Tangents

Sum and Difference Formulas With Inverse Functions

Double-angle and Half-angle Formulas

Section 6.5

Double-angle Formulas

Theorem. [Double-angle Formulas]

sin(2µ) = 2sinµ cosµcos(2µ) = cos2µ { sin2µ

cos(2µ) = 1 { 2sin2µcos(2µ) = 2cos2µ { 1

Double-angle Formulas

Example. If , find the exact values.

(a) Problem: sin(2µ)

Answer:

(b) Problem: cos(2µ)

Answer:

Identities using Double-angle Formulas

Double-angle Formula for Tangent

Formulas for Squares

Identities using Double-angle Formulas

Example. An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. A first approximation to the sawtooth curve is given by

Show thaty = sin(2¼x)cos2(¼x)

Identities using Double-angle Formulas

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

Half-angle Formulas Theorem. [Half-angle Formulas]

where the + or { sign is determined by the quadrant of the angle

Half-angle Formulas

Example. Use a half-angle formula to find the exact value of

(a) Problem: sin 15±

Answer:

(b) Problem:

Answer:

Half-angle Formulas

Example. If , find the exact values.

(a) Problem:

Answer:

(b) Problem:

Answer:

Half-angle Formulas

Alternate Half-angle Formulas for Tangent

Key Points

Double-angle Formulas Identities using Double-

angle FormulasHalf-angle Formulas

Product-to-Sum and Sum-to-Product Formulas

Section 6.6

Product-to-Sum Formulas

Theorem. [Product-to-Sum Formulas]

Product-to-Sum Formulas

Example. Express each of the following products as a sum containing only sines or cosines(a) Problem: cos(4µ)cos(2µ)

Answer:(b) Problem: sin(3µ)sin(5µ)

Answer:(c) Problem: sin(4µ)cos(6µ)

Answer:

Sum-to-Product Formulas

Theorem. [Sum-to-Product Formulas]

Sum-to-Product Formulas

Example. Express each sum or difference as a product of sines and/or cosines

(a) Problem: sin(4µ) + sin(2µ)Answer:

(b) Problem: cos(5µ) { cos(3µ)Answer:

Key Points

Product-to-Sum FormulasSum-to-Product Formulas

Trigonometric Equations (I)

Section 6.7

Trigonometric Equations

Trigonometric Equations: Equations involving trigonometric functions that are satisfied by only some or no values of the variable

Values satisfying the equation are the solutions of the equation

IMPORTANT! Identities are different

Every value in the domain satisfies an identity

Checking Solutions of Trigonometric

Equations Example. Determine whether

the following are solutions of the equation

(a) Problem:

Answer:

(b) Problem:

Answer:

Solving Trigonometric Equations

Example. Solve the equations. Give a general formula for all the solutions.

(a) Problem:

Answer:

(b) Problem:

Answer:

Solving Trigonometric Equations

Example. Solve the equations on the interval 0 · x < 2¼.

(a) Problem:

Answer:

(b) Problem:

Answer:

Approximating Solutions to

Trigonometric Equations Example. Use a calculator to

solve the equations on the interval 0 · x < 2¼. Express answers in radians, rounded to two decimal places.

(a) Problem: tan µ = 4.2

Answer:

(b) Problem: 2 csc µ = 5

Answer:

Key Points

Trigonometric EquationsChecking Solutions of

Trigonometric EquationsSolving Trigonometric

EquationsApproximating Solutions to

Trigonometric Equations

Trigonometric Equations (II)

Section 6.8

Solving Trigonometric Equations Quadratic in

FormExample. Solve the

equations on the interval 0 · x < 2¼.

(a) Problem:

Answer:

(b) Problem:

Answer:

Solving Trigonometric Equations Using

IdentitiesExample. Solve the

equations on the interval 0 · x < 2¼.

(a) Problem:

Answer:

(b) Problem:

Answer:

Trigonometric Equations Linear in Sine

and CosineExample. Solve the

equations on the interval 0 · x < 2¼.

(a) Problem:

Answer:

(b) Problem:

Answer:

Trigonometric Equations Using a Graphing Utility

Example. Problem: Use a calculator to

solve the equation2 + 13sin x = 14cos2 x

on the interval 0 · x < 2¼. Express answers in degrees, rounded to one decimal place.

Answer:

Trigonometric Equations Using a Graphing Utility

Example. Problem: Use a calculator to

solve the equation2x { 3cos x = 0

on the interval 0 · x < 2¼. Express answers in radians, rounded to two decimal places.

Answer:

Key Points

Solving Trigonometric Equations Quadratic in Form

Solving Trigonometric Equations Using Identities

Trigonometric Equations Linear in Sine and Cosine

Trigonometric Equations Using a Graphing Utility