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7
Trigonometric Identities and Equations
© 2008 Pearson Addison-Wesley.All rights reserved
Sections 7.5–7.7
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7.5Inverse Circular Functions7.6Trigonometric Equations7.7
Equations Involving Inverse Trigonometric Functions
Trigonometric Identities and Equations7
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Inverse Circular Functions7.5Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular Functions ▪ Inverse Function Values
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Find y in each equation.
7.5 Example 1 Finding Inverse Sine Values (page 688)
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7.5 Example 1 Finding Inverse Sine Values (cont.)
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7.5 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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Find y in each equation.
7.5 Example 2 Finding Inverse Cosine Values (page 689)
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Find y in each equation.
7.5 Example 2 Finding Inverse Cosine Values (page 689)
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7.5 Example 3 Finding Inverse Function Values (Degree-Measured Angles) (page 692)
Find the degree measure of θ in each of the following.
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7.5 Example 4 Finding Inverse Function Values With a Calculator (page 693)
(a) Find y in radians if
With the calculator in radian mode, enter as
y = 1.823476582
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7.5 Example 4(b) Finding Inverse Function Values With a Calculator (page 693)
(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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7.5 Example 4(b) Finding Inverse Function Values With a Calculator (cont.)
θ = 104.1871349°
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7.5 Example 5 Finding Function Values Using Definitions of the Trigonometric Functions (page 693)
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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7.5 Example 5(a) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)
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7.5 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (page 693)
Since arccot is defined only in quadrants I and II, and
is negative, θ is in quadrant
II.
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7.5 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)
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7.5 Example 6(a) Finding Function Values Using Identities
(page 694) Evaluate the expression without a calculator.
Use the cosine difference identity:
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7.5 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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7.5 Example 6(a) Finding Function Values Using Identities
(cont.)
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7.5 Example 6(b) Finding Function Values Using Identities
(page 694) Evaluate the expression without a calculator.
Use the double-angle sine identity:
sin(2 arccot (–5))
Let A = arccot (–5), so cot A = –5.
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7.5 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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7.5 Example 6(b) Finding Function Values Using Identities
(cont.)
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7.5 Example 7(a) Finding Function Values in Terms of u
(page 695) Write , as an algebraic expression in u.
Sketch θ in quadrant I. Use the Pythagorean theorem to find the missing side.
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7.5 Example 7(a) Finding Function Values in Terms of u
(cont.)
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7.5 Example 7(b) Finding Function Values in Terms of u
(page 695) Write , u > 0, as an algebraic expression in u.
Sketch θ in quadrant I. Use the Pythagorean theorem to find the missing side.
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7.5 Example 7(b) Finding Function Values in Terms of u
(cont.)
Use the double-angle sine identity to find sin 2θ.
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7.5 Example 8 Finding the Optimal Angle of Elevation of a Shot Put (page 696)
The optimal angle of elevation θ a shot-putter should aim for to throw the greatest distance depends on the velocity v and the initial height h of the shot.
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7.5 Example 8 Finding the Optimal Angle of Elevation of a Shot Put (cont.)
Suppose a shot-putter can consistently throw the steel ball with h = 7.5 ft and v = 50 ft per sec. At what angle should he throw the ball to maximize distance?
One model for θ that achieves this greatest distance is
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Trigonometric Equations7.6Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities ▪ Equations with Half-Angles ▪ Equations with Multiple Angles ▪ Applications
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7.6 Example 1 Solving a Trigonometric Equation by Linear Methods (page 701)
is positive in quadrants I and III.
The reference angle is 30° because
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7.6 Example 1 Solving a Trigonometric Equation by Linear Methods (cont.)
Solution set: {30°, 210°}
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7.6 Example 2 Solving a Trigonometric Equation by Factoring (page 701)
or
Solution set: {90°, 135°, 270°, 315°}
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7.6 Example 3 Solving a Trigonometric Equation by Factoring (page 702)
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7.6 Example 3 Solving a Trigonometric Equation by Factoring (cont.)
has one solution,
has two solutions, the angles in quadrants III and IV with the reference angle .729728:3.8713 and 5.5535.
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7.6 Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (page 702)
Find all solutions of
Use the quadratic formula with a = 1, b = 2, and c = –1 to solve for cos x.
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7.6 Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (cont.)
Since there are two solutions, one in quadrant I and the other in quadrant IV.
Since , there are no solutions for this value of cos x.
To find all solutions, add integer multiples of the period of cosine, 2
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7.6 Example 5 Solving a Trigonometric Equation by Squaring (page 703)
Square both sides.
The possible solutions are
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7.6 Example 5 Solving a Trigonometric Equation by Squaring (cont.)
Since the solution was found by squaring both sides of an equation, we must check that each proposed solution is a solution of the original equation.
Not a solution Solution
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7.6 Example 6 Solving an Equation Using a Half-Angle Identity (page 704)
(a) over the interval and (b) give all solutions.
is not in the requested domain.
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7.6 Example 6 Solving an Equation Using a Half-Angle Identity (cont.)
This is a cosine curve with period
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7.6 Example 7 Solving an Equation With a Double Angle
(page 705)
Factor.
or
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7.6 Example 8 Solving an Equation Using a Multiple Angle Identity (page 705)
From the given interval 0° ≤ θ < 360°, the interval for 2θ is 0° ≤ 2θ < 720°.
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7.6 Example 8 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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7.6 Example 9 Describing a Musical Tone From a Graph
(page 706) A basic component of music is a pure tone. The graph below models the sinusoidal pressure y = P in pounds per square foot from a pure tone at time x = t in seconds.
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7.6 Example 9(a) Describing a Musical Tone From a Graph
(page 706) The frequency of a pure tone is often measured in hertz. One hertz is equal to one cycle per second and is abbreviated Hz. What is the frequency f in hertz of the pure tone shown in the graph?
There are 4 cycles in .0182 seconds.
The frequency is 220 Hz.
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7.6 Example 9(b) Describing a Musical Tone From a Graph
(page 706) The time for the tone to produce one complete cycle is called the period.
Approximate the period T in seconds of the pure tone.
Four periods cover a time of .0182 seconds.
One period =
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7.6 Example 9(c) Describing a Musical Tone From a Graph
(page 706) Use a calculator to estimate the first solution to the equation that makes y = .002 over the interval [0, .0182].
The first point of intersection is at about x = .00053 sec.
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7.6 Example 10 Analyzing Pressures of Upper Harmonics
(page 707) Suppose that the E key above middle C is played on a piano. Its fundamental frequency isand its associate pressure is expressed as
(a) What are the next four frequencies at which the string will vibrate?
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7.6 Example 10 Analyzing Pressures of Upper Harmonics
(cont.) (b) What are the pressures corresponding to these
four upper harmonics?
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Equations Involving Inverse Trigonometric Functions 7.7Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations
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7.7 Example 1 Solving an Equation for a Variable Using Inverse Notation (page 713)
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7.7 Example 2 Solving an Equation Involving an Inverse Trigonometric Function (page 713)
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7.7 Example 3 Solving an Equation Involving Inverse Trigonometric Functions (page 714)
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7.7 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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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (page 714)
Isolate one inverse function on one side of the equation:
Sine difference identity
By definition, the arcsine function is defined in
quadrants I and IV, so
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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)
From equation (1),
Sketch u in Quadrant III. Use the Pythagorean theorem to find the missing side.
By definition, the range of arccos x is so the intersection of the two ranges is
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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)
Substitute into equation (2):
Square both sides.
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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)
Check each potential solution.
There is no value of x in the given domain
such that
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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)
Range of arcsine is
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7.7 Example 4 Solving an Inverse Trigonometric Equation Using an Identity (cont.)