Transcript Chapter 5

5
Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
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5.1 Trigonometric Identities
Fundamental Identities ▪ Using the Fundamental Identities
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5.1
Example 1 Finding Trigonometric Function Values Given
One Value and the Quadrant
If
value.
and
is in quadrant IV, find each function
(a)
In quadrant IV,
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is negative.
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5.1
Example 1 Finding Trigonometric Function Values Given
One Value and the Quadrant (cont.)
If
value.
and
is in quadrant IV, find each function
(b)
(c)
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5.1
Example 2 Expressing One Function in Terms of Another
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5.1
Example 3 Rewriting an Expression in Terms of Sine and
Cosine
Write
in terms of
then simplify the expression.
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and
, and
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5.2 Verifying Trigonometric Identities
Verifying Identities by Working With One Side ▪ Verifying
Identities by Working With Both Sides
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5.2
Example 1 Verifying an Identity (Working With One Side)
Verify that
is an identity.
Left side of given
equation
Right side of given equation
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5.2
Example 2 Verifying an Identity (Working With One Side)
Verify that
is an identity.
Simplify.
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5.2
Example 3 Verifying an Identity (Working With One Side)
Verify that
is an identity.
Simplify.
Factor.
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5.2
Example 4 Verifying an Identity (Working With One Side)
Verify that
is an identity.
Multiply by 1
in the form
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5.2
Example 5 Verifying an Identity (Working With Both
Sides)
Verify that
identity.
is an
Working with the left side:
Multiply by 1 in the form
Simplify.
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5.2
Example 5 Verifying an Identity (Working With Both
Sides) (cont.)
Working with the right side:
Factor the numerator.
Distributive property.
Factor the
denominator.
Simplify.
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5.2
Example 5 Verifying an Identity (Working With Both
Sides) (cont.)
So, the identity is verified.
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5.3
Sum and Difference Identities for
Cosine
Difference Identity for Cosine ▪ Sum Identity for Cosine ▪
Cofunction Identities ▪ Applying the Sum and Difference
Identities
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5.3
Example 1 Finding Exact Cosine Function Values
Find the exact value of each expression.
(a)
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5.3
Example 1 Finding Exact Cosine Function Values
(cont.)
(b)
(c)
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5.3
Example 2 Using Cofunction Identities to Find θ
Find an angle θ that satisfies each of the following.
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5.3
Example 3 Reducing cos (A – B) to a Function of a
Single Variable
Write cos(90° + θ) as a trigonometric function of θ
alone.
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5.3
Example 2B Using Cofunction Identities to Find θ
(Miscellaneous HW Examples)
Find an angle θ that satisfies each of the following.
1. Sin (θ + 15o) = Cos (2θ + 5o)
Now see 5.3 # 38
2. Write Cos π/12 as cofunction
Now see 5.3 #18
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t
Suppose that
, and both s and t
are in quadrant IV. Find cos(s – t).
The Pythagorean theorem gives
Since s is in quadrant IV, y = –8.
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t (cont.)
Use a Pythagorean identity to find the value of cos t.
Since t is in quadrant IV,
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5.3
Example 4 Finding cos (s + t) Given Information About
s and t (cont.)
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5.4
Sum and Difference Identities for
Sine and Tangent
Sum and Difference Identities for Sine ▪ Sum and Difference
Identities for Tangent ▪ Applying the Sum and Difference
Identities
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values
Find the exact value of each expression.
(a)
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values (cont.)
(b)
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5.4
Example 1 Finding Exact Sine and Tangent Function
Values (cont.)
(c)
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5.4
Example 2 Writing Functions as Expressions Involving
Functions of θ
Write each function as an expression involving
functions of θ.
(a)
(b)
(c)
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5.4
Example 4 Verifying an Identity Using Sum and
Difference Identities
Verify that the equation is an identity.
Combine the fractions.
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5.4
Example 4 Verifying an Identity Using Sum and
Difference Identities (cont.)
Expand the terms.
Combine terms.
Factor.
So, the identity is verified.
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5.5 Double-Angle Identities
Double-Angle Identities ▪ Omit Product-to-Sum and Sum-toProduct Identities
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ
The identity for sin 2θ requires cos θ.
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ (cont.)
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5.5
Example 1 Finding Function Values of 2θ Given
Information About θ (cont.)
Alternatively, find tan θ and then use the tangent
double-angle identity.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ
Find the values of the six trigonometric functions of θ
if
Use the identity
to find sin θ:
θ is in quadrant III, so sin θ is negative.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ (cont.)
Use the identity
to find cos θ:
θ is in quadrant III, so cos θ is negative.
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5.5
Example 2 Finding Function Values of θ Given
Information About 2θ (cont.)
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5.6 Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities
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5.6
Example 1 Using a Half-Angle Identity to Find an Exact
Value
Find the exact value of sin 22.5° using the half-angle
identity for sine.
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5.6
Example 2 Using a Half-Angle Identity to Find an Exact
Value
Find the exact value of tan 75° using the identity
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5.6 Example 3
Finding Function Values of s Given
2
Information About s
The angle associated with
is positive while
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lies in quadrant II since
are negative.
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5.6 Example 3
Finding Function Values of s Given
2
Information About s (cont.)
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6
Inverse Circular Functions and Trigonometric
Equations
6.1 Inverse Circular Functions
6.2 Trigonometric Equations I
6.3 Trigonometric Equations II
6.4 Equations Involving Inverse
Trigonometric Functions
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6.1 Inverse Circular Functions
Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine
Function ▪ Inverse Tangent Function ▪ Remaining Inverse
Circular Functions ▪ Inverse Function Values
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6.1
Example 1 Finding Inverse Sine Values
Find y in each equation.
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6.1
Example 1 Finding Inverse Sine Values (cont.)
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6.1
Example 1 Finding Inverse Sine Values (cont.)
is not in the domain of the inverse sine function,
[–1, 1], so
does not exist.
A graphing calculator will give an error message for
this input.
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6.1
Example 2 Finding Inverse Cosine Values
Find y in each equation.
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6.1
Example 2 Finding Inverse Cosine Values
Find y in each equation.
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6.1
Example 3 Finding Inverse Function Values (DegreeMeasured Angles)
Find the degree measure of θ in each of the following.
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6.1
Example 4 Finding Inverse Function Values With a
Calculator
(a) Find y in radians if
With the calculator in radian mode, enter
as
y = 1.823476582
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6.1
Example 4(b) Finding Inverse Function Values With a
Calculator
(b) Find θ in degrees if θ = arccot(–.2528).
A calculator gives the inverse cotangent value of a
negative number as a quadrant IV angle.
The restriction on the range of arccotangent implies
that the angle must be in quadrant II, so, with the
calculator in degree mode, enter arccot(–.2528) as
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6.1
Example 4(b) Finding Inverse Function Values With a
Calculator (cont.)
θ = 104.1871349°
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6.1
Example 5 Finding Function Values Using Definitions of
the Trigonometric Functions
Evaluate each expression without a calculator.
Since arcsin is defined only in quadrants I and IV, and
is positive, θ is in quadrant I.
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6.1
Example 5(a) Finding Function Values Using Definitions
of the Trigonometric Functions (cont.)
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6.1
Example 5(b) Finding Function Values Using Definitions
of the Trigonometric Functions
Since arccot is defined only in quadrants I and II, and
is negative, θ is in quadrant II.
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6.1
Example 5(b) Finding Function Values Using Definitions
of the Trigonometric Functions (cont.)
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6.1
Example 6(a) Finding Function Values Using Identities
Evaluate the expression without a calculator.
Use the cosine difference identity:
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6.1
Example 6(a) Finding Function Values Using Identities
(cont.)
Sketch both A and B in quadrant I. Use the
Pythagorean theorem to find the missing side.
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6.1
Example 6(a) Finding Function Values Using Identities
(cont.)
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6.1
Example 6(b) Finding Function Values Using Identities
Evaluate the expression without a calculator.
sin(2 arccot (–5))
Let A = arccot (–5), so cot A = –5.
Use the double-angle sine identity:
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6.1
Example 6(b) Finding Function Values Using Identities
(cont.)
Sketch A in quadrant II. Use the Pythagorean
theorem to find the missing side.
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6.1
Example 6(b) Finding Function Values Using Identities
(cont.)
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6.2 Trigonometric Equations I
Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by
Quadratic Methods ▪ Solving by Using Trigonometric Identities
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6.2
Example 1 Solving a Trigonometric Equation by Linear
Methods
is positive in quadrants I and III.
The reference angle is 30° because
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6.2
Example 1 Solving a Trigonometric Equation by Linear
Methods (cont.)
Solution set: {30°, 210°}
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6.2
Example 2 Solving a Trigonometric Equation by
Factoring
or
Solution set: {90°, 135°, 270°, 315°}
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6.2
Example 3 Solving a Trigonometric Equation by
Factoring
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6.2
Example 3 Solving a Trigonometric Equation by
Factoring (cont.)
has
two solutions, the angles in
quadrants III and IV with the
reference angle .729728:
3.8713 and 5.5535.
has one solution,
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6.3 Trigonometric Equations II
Equations with Half-Angles ▪ Equations with Multiple Angles
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6.3
Example 1 Solving an Equation Using a Half-Angle
Identity
(a) over the interval
and (b) give all solutions.
is not in the requested domain.
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6.3
Example 2 Solving an Equation With a Double Angle
Factor.
or
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6.3
Example 3 Solving an Equation Using a Multiple Angle
Identity
From the given interval 0° ≤ θ < 360°, the interval for
2θ is 0° ≤ 2θ < 720°.
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6.3
Example 3 Solving an Equation Using a Multiple Angle
Identity (cont.)
Since cosine is negative in quadrants II and III,
solutions over this interval are
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6.4
Equations Involving Inverse
Trigonometric Functions
Solving for x in Terms of y Using Inverse Functions ▪ Solving
Inverse Trigonometric Equations
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6.4
Example 1 Solving an Equation for a Variable Using
Inverse Notation
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6.4
Example 2 Solving an Equation Involving an Inverse
Trigonometric Function
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6.4
Example 3 Solving an Equation Involving Inverse
Trigonometric Functions
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6.4
Example 3 Solving an Equation Involving Inverse
Trigonometric Functions (cont.)
Sketch u in quadrant I. Use
the Pythagorean theorem to
find the missing side.
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