Download Trigonometric Functions

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 oneto-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
Inverse Sine Function


The sine function is not one-to-one
We restrict to domain
4
2
3
2
3
2
2
-2
-4
2
Inverse Sine Function

Inverse sine function: Inverse of the
domain-restricted sine function
3
2
2
-4
-2
2
2
3
2
4
Inverse Sine Function
y = sin{1x means x = sin y
 Must have {1 · x · 1 and
 Many books write y = arcsin x
WARNING!



The {1 is not an exponent, but an
indication of an inverse function
Domain is {1 · x · 1
Range 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-one
We restrict to domain [0, ¼]
4
2
3
2
3
2
2
-2
-4
2
Inverse Cosine Function

Inverse cosine function: Inverse of
the domain-restricted cosine function
3
2
2
-4
-2
2
2
3
2
4
Inverse Cosine Function




y = cos{1x means x = cos y
Must have {1 · x · 1 and 0 · y · ¼
Can also write y = arccos x
Domain is {1 · x · 1
Range 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
6
4
2
2
3
2
3
2
2
-2
-4
-6
2
2
Inverse Tangent Function

Inverse tangent function: Inverse of
the domain-restricted tangent
function
3
2
2
-4
-2
2
2
3
2
4
Inverse Tangent Function




y = tan{1x means x = tan y
Have {1 · x · 1 and
Also write y = arctan x
Domain is all real numbers
Range is
Exact Values of the Inverse
Tangent Function

Example. Find the exact values of:
(a) Problem:
Answer:
(b) Problem:
Answer:
The Inverse
Trigonometric
Functions
[Continued]
Section 6.2
Exact Values Involving Inverse
Trigonometric Functions

Example. Find the exact values of the
following expressions
(a) Problem:
Answer:
(b) Problem:
Answer:
Exact Values Involving Inverse
Trigonometric Functions

Example. 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
6
3
2
4
2
2
2
3
2
3
2
2
2
-6
-4
-2
2
2
2
-2
-4
3
2
-6
2
4
6
Inverse Secant, Cosecant and
Cotangent Functions

Inverse Cosecant Function
y = csc{1x means x = csc y

y0
j x j ¸ 1,
2
6
3
2
4
2
2
2
3
2
3
2
2
2
-6
-4
-2
2
2
2
-2
-4
3
2
-6
2
4
6
Inverse Secant, Cosecant and
Cotangent Functions
Inverse Cotangent Function
y = cot{1x means x = cot y


{1 < x < 1, 0 < y < ¼
2
6
3
2
4
2
2
2
3
2
3
2
2
2
-6
-4
-2
2
2
2
-2
-4
3
2
-6
2
4
6
Inverse Secant, Cosecant
and Cotangent Functions

Example. 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
Identities

Quotient 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





Identities
Fundamental Trigonometric Identities
Simplifying Using Identities
Establishing Identities
Guidelines 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
4
4
2
2
-2
2
4
-4
-2
2
-2
-2
-4
-4
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 Functions

Example. Find the exact value of
each expression
(a) Problem:
Answer:
(b) Problem:
Answer:
Sum and Difference Formulas
With Inverse Functions

Example. 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 cosµ
cos(2µ) = cos2µ { sin2µ
cos(2µ) = 1 { 2sin2µ
cos(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 that
y = sin(2¼x)cos2(¼x)
Identities using Double-angle
Formulas
-4
4
4
2
2
-2
2
4
-4
-2
2
-2
-2
-4
-4
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
the exact values.
(a) Problem:
Answer:
(b) Problem:
Answer:
, find
Half-angle Formulas

Alternate Half-angle Formulas for
Tangent
Key Points



Double-angle Formulas
Identities using Double-angle
Formulas
Half-angle Formulas
Product-to-Sum
and Sum-toProduct 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 Formulas
Sum-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 Equations
Checking Solutions of Trigonometric
Equations
Solving Trigonometric Equations
Approximating Solutions to
Trigonometric Equations
Trigonometric
Equations (II)
Section 6.8
Solving Trigonometric
Equations Quadratic in Form

Example. Solve the equations on the
interval 0 · x < 2¼.
(a) Problem:
Answer:
(b) Problem:
Answer:
Solving Trigonometric
Equations Using Identities

Example. Solve the equations on the
interval 0 · x < 2¼.
(a) Problem:
Answer:
(b) Problem:
Answer:
Trigonometric Equations
Linear in Sine and Cosine

Example. 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
equation
2 + 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
equation
2x { 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