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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
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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
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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]
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Inverse Cosine Function
Inverse cosine function: Inverse of the domain-restricted cosine function
-4 -2 2 4
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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
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Inverse Tangent Function
Inverse tangent function: Inverse of the domain-restricted tangent function
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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,
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Inverse Secant, Cosecant and
Cotangent Functions Inverse Cosecant Function
y = csc{1x means x = csc y j x j ¸ 1, y 0
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2
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-6 -4 -2 2 4 6
2
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2
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Inverse Secant, Cosecant and
Cotangent Functions Inverse Cotangent Function
y = cot{1x means x = cot y{1 < x < 1, 0 < y < ¼
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-6 -4 -2 2 4 6
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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
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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
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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