1. Inverse Circular Functions
6.1
6.2
6.3
6.4
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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2. Restricted Domain and Ranges so that the Inverses of Trigonometric Functions will be Functions
On the white board we will discuss the basic sine graph and its inverse ….
What restrictions need to be made so that the sin-1 graph is a function?
Inverse Sine, Cosecant, and Tangent Functions
are restricted to quad I & IV
Inverse Cosine, Secant, and Cotangent Functions
are restricted to quad I & II
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3. 6.1 Example 1 Finding Inverse Sine Values (page 261)
Find y in each equation.
Find reference angle: Sin of what angle is ½? The angle will be in quadrant IV
sin of what angle is ?
y = 60˚ = π/3
Ref angle = 30˚ = π/6
y = -30˚ = -π/6
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4. 6.1 Example 1 Finding Inverse Sine Values (page 261)
Find y in each equation.
is not in the domain of the inverse sine function, [–1, 1], so does not exist.
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5. 6.1 Example 2 Finding Inverse Cosine Values (page 262)
Find y in each equation.
0 < y < π
cos y = 0
0 < y < π
cos y = ½
The cos of what angle is 0?
The cos of what angle is ½?
y = 90˚ = π/2
y = 60˚ = π/3
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6. Find the degree measure of θ in each of the following.
6.1 Example 3 Finding Inverse Function Values (Degree-Measured Angles) (page 265)
Find the degree measure of θ in each of the following.
tan of what angle is ?
Reference angle = 45˚ Quad IV
y = 60˚ = π/3
y = –45˚ = –π/4
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7. 6.1 Ex 4 Finding Inverse Function Values With a Calculator (pg 266)
(a) Find y in radians if
(b) Find θ in degrees if θ = arccot(–.2528).
0 < θ < 180˚
0 < y < π
cot θ = –.2528
tan θ = –1/.2528
θ = tan-1 (–1/.2528)
Ref θ = – ˚
sec y = – 4
cos y = (–1/4)
y = cos-1 (–1/4)
y =
θ = °
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8. Evaluate each expression without a calculator.
6.1 Example 5 Finding Function Values Using Definitions of the Trigonometric Functions (page 266)
Evaluate each expression without a calculator.
Let θ = sin-1 (2/3)
Trying to find cos (θ) =
sin-1 restricted to quad I & IV
sin-1 is positive (2/3)
reference angle will be in quad I
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9. cot-1 restricted to quad I & II cot-1 is negative (15/8)
6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (page 266)
Let θ = cot-1 (–15/8)
Trying to find sec (θ) =
cot-1 restricted to quad I & II
cot-1 is negative (15/8)
reference angle will be in quad II
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10. 6.1 Example 6(a) Finding Function Values Using Identities (page 267)
Evaluate the expression without a calculator.
Let A = arctan 4/3 and B = arccos 12/13
= sin (A – B)
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11. 6.1 Ex 6(b) Finding Function Values Using Identities (pg 267)
Evaluate the expression without a calculator.
sin(2 arccot (–5))
Let A = arccot (-5) cot-1 restricted to quad I & II reference angle quad II
= sin(2A)
= 2 sinA cosA
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12. 6.1 Example 7(a) Finding Function Values in Terms of u (page 268)
Write , as an algebraic expression in u.
Let θ = sec-1u
secθ = u
cos θ = 1/u
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13. 6.1 Example 7(b) Finding Function Values in Terms of u (page 268)
Write , u > 0, as an algebraic expression in u.
Let θ = cos-1u
cos θ = u
Quad I
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14. 6.1 Example 8 Finding the Optimal Angle of Elevation of a Shot Put (page 269)
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.
One model for θ that achieves this greatest distance is
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?

15. Inverse Circular Functions
6.1
Inverse Circular Functions
Inverse Sine, Cosecant, and Tangent Functions
are restricted to quad I & IV
Inverse Cosine, Secant, and Cotangent Functions
are restricted to quad I & II
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16. Trigonometric Equations I
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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17. 6.2 Introduction Solve for θ? Quad: Ref. angle:
Quad:
Ref. angle:
Answer between 0⁰ and 360⁰
Answer between 0 and 2π
All answers in degrees
All answers in radians
I and II
30⁰
30⁰and 150⁰
π/6 and 5π/6
30⁰+360⁰n, 150⁰+360⁰n where n ε Z
π/6 + 2πn, 5π/6 + 2πn where n ε Z
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18. 6.2 Introduction Solve for θ? Quad: Ref. angle:
Quad:
Ref. angle:
Answer between 0⁰ and 360⁰
Answer between 0 and 2π
All answers in degrees
All answers in radians
I and IV
30⁰
30⁰and 330⁰
π/6 and 11π/6
30⁰+360⁰n, 330⁰+360⁰n where n ε Z
π/6 + 2πn, 11π/6 + 2πn where n ε Z
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19. 6.2 Introduction Solve for θ? Quad: Ref. angle:
Quad:
Ref. angle:
Answer between 0⁰ and 360⁰
Answer between 0 and 2π
All answers in degrees
All answers in radians
II and IV
60⁰
120⁰and 300⁰
2π/3 and 5π/3
120⁰+360⁰n, 300⁰+360⁰n where n ε Z
2π/3 + 2πn, 5π/3 + 2πn where n ε Z
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20. is positive in quadrants I and III.
6.2 Example 1 Solving a Trigonometric Equation by Linear Methods (page 274)
is positive in quadrants I and III.
The reference angle is 30°
Solution set: {30°, 210°}
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21. 6.2 Example 2 Solving a Trigonometric Equation by Factoring (page 274)
Cot neg in Quad II & IV
Ref angle = 45⁰
Where is the x-corr on the axis = 0? or
Solution set: {90°, 135°, 270°, 315°}
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22. 6.2 Example 3 Solving a Trigonometric Equation by Factoring (page 275)
3u2 – u – 2 = 0
3u2 – 3u + 2u – 2 = 0
3u(u – 1) + 2(u – 1) = 0
(3u + 2)(u – 1) = 0
u = -2/3 or u = 1
sin x = -2/3 will have a ref angle of
Answer in quad III & IV
π , and 2π –
sin x = 1
When is the y-corr 1 on the axis of the unit circle?
sin x = -2/3 or sin x = 1
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23. 6.2 Example 4 Solving a Trigonometric Equation Using the Quadratic Formula (page 275)
Find all solutions of
(answer in radians)
Use the quadratic formula with a = 1, b = 2, and c = –1 to solve for cos x.
Get the program from me. At this time you will be allowed to use Quad Program
cos x
cos x = or cos x =
cos x = will have a reference angle of and the answers will be in quad I and IV and the multiples of 2π …
cos x = is NOT possible
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24. 6.2 Example 5 Solving a Trigonometric Equation by Squaring (page 276)
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
When tan x = √3? Ref angle = 60˚ = π/3 and results will be in quad I and III
The possible solutions are
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Solution

25. 6.2 Example 6 Describing a Musical Tone From a Graph (page 277)
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.
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 seconds.
The time for the tone to produce one complete cycle is called the period. Approximate the period T in seconds of the pure tone.
The frequency is 220 Hz.
Period 2π/440π = 1/220
Four periods cover a time of seconds.
One period =
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26. 6.2 Summary Solving Trigonometric Functions: Solving by Linear Methods
Solving by Factoring
Solving by Quadratic Methods
Solving by Using Trigonometric Identities
Reminder: All Students Take Calculus
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27. Trigonometric Equations II
6.3
Trigonometric Equations II
Equations with Half-Angles ▪ Equations with Multiple Angles
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28. 6.3 Ex 1 Solving an Equation Using a Half-Angle Identity (pg 280)
(a) over the interval and (b) give all solutions.
π/4
Reference angle
Quad I & IV, 45˚= π/4
7π/4
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29. 6.3 Ex 2 Solving an Equation With a Double Angle (pg 281)
5π/6
π/6
3π/2 or
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30. 6.3 Ex 3 Solving an Equation Using a Multiple Angle Identity (pg 282)
Reference angle
Quad II & III 60°.
Angles 120˚ and 240˚
2θ = 120˚ + 360˚n & 2θ = 240˚ + 360˚n
θ = 60˚ + 180˚n θ = 120˚ + 180˚n
θ = 60˚, 240˚, 420˚,… θ = 120˚, 300˚, 480˚,…
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31. 6.3 Ex 4 Solving an Equation With a Multiple Angle (page 282)
π/2
y/x
cot 2x = 0
tan 2x = undefined
3π/2
2x = π/2, 3π/2, 5π/2, 7π/2, 9π/2, 11π/2,…
x = π/4, 3π/4, 5π/4, 7π/4, 9π/4, 11π/4, …
Because you squared both sides you must check the answers
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32. 6.3 Ex 4 Solving an Equation With a Multiple Angle (cont.)☺ ☺
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33. 6.3 Ex 5 Analyzing Pressures of Upper Harmonics (page 283)
Suppose that the E key above middle C is played on a piano. Its fundamental frequency is and its associate pressure is expressed as
What are the next four frequencies at which the string will vibrate?
Frequency = the reciprocal of the period
Period = 2π/b Frequency = b/2π = 660π/2π = 330 Hz
f2 = f = 660 Hz
f3 = f = 990 Hz
f4 = f = 1320 Hz
f5 = f = 1650 Hz
The string will vibrate at the next four frequencies: 660 Hz, 990 Hz, 1320 Hz, & 1650 Hz
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34. 6.3 Summary Solving More Trigonometric Equations:
Still need “All Students Take Calculus”
If you raise both sides to an even power, you must check your solutions.
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35. Equations Involving Inverse Trigonometric Functions
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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36. 6.4 Example 1 Solving an Equation for a Variable Using Inverse Notation (page 287)
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37. 6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions (page 288)
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38. Solve cos-1x = sin-1(3/5) x = cos(sin-1(3/5)) x = 4/5 6.4 add on 5 3 4
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39. 6.4 add on
Solve
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40. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.
6.4 add on
Solve

41. 6.4 add on Use a Graphing Calculator to Solve
Solve (arctan x)4 – 2x + 6 = 0, in the interval [0,6]
y1 = (tan-1(x))4 – 2x + 6
y2 = 0
Adjust the window for the domain [0,6]
Answer: x ≈
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42. 6.4 Summary
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