Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 - - PDF document
Slide 1 / 207 Slide 2 / 207 Pre-Calc Trigonometry 2015-03-24 www.njctl.org Slide 3 / 207 Table of Contents click on the topic to go Unit Circle to that section Graphing Law of Sines Law of Cosines Pythagorean Identities Angle
www.njctl.org 2015-03-24
Table of Contents
Unit Circle Graphing Law of Sines Law of Cosines Pythagorean Identities Angle Sum/Difference Double Angle Half Angle Sum to Product Inverse Trig Functions Trig Equations Product to Sum Power Reducing
click on the topic to go to that section
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Unit Circle
Students will understand how to use the Unit Circle to find angles and determine their trigonometric value.
The Unit Circle is a tool that allows us to determine the location of any angle. Unit Circle
Special Right Triangles
Unit Circle
Example 1: Find a Example 2: Find b & c 6 a 4 b c
Unit Circle
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Example 3: Find d Example 4: Find e 8 d e 9
Unit Circle
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Example 5: Find f Example 6: Find g & h 1 f 1 h g
Unit Circle
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30o 45o 60o 30o 45o 60o 30o 45o 60o 30o 45o 60o
Unit Circle
Unit Circle
Unit Circle
Unit Circle
Unit Circle
4 Which function is positive in the second quadrant? Choose all that apply.
A
cos x
B
sin x
C
tan x
D
sec x
E
csc x
F
cot x
Unit Circle
Teacher Teacher
5 Which function is positive in the fourth quadrant? Choose all that apply.
A
cos x
B
sin x
C
tan x
D
sec x
E
csc x
F
cot x
Unit Circle
Teacher Teacher
6 Which function is positive in the third quadrant? Choose all that apply.
A
cos x
B
sin x
C
tan x
D
sec x
E
csc x
F
cot x
Unit Circle
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Teacher Teacher
Example: Given the terminal point of (
find sin x, cos x, and tan x.
Unit Circle
7 Given the terminal point find tan x.
Unit Circle
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8 Given the terminal point find sin x.
Unit Circle
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9 Given the terminal point find tan x.
Unit Circle
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10 Knowing sin x = Find cos x if the terminal point is in the first quadrant
Unit Circle
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11 Knowing sin x = Find cos x if the terminal point is in the 2
nd quadrant
Unit Circle
Teacher Teacher
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Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
Graphing
Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
Graphing
Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
Graphing
Graphing cos, sin, & tan Graph by using values from the table. Since the values are based on a circle, values will repeat.
Graphing
x cos x
Amplitude Period
Graphing
y= a sin(x) or y= a cos(x) In the study of transforming parent functions, we learned "a" was a vertical stretch or shrink. For trig functions it is called the amplitude.
Graphing
In y= cos(x), a=1 This means at any time, y= cos (x) is at most 1 away from the axis it is oscillating about. Find the amplitude: y= 3 sin(x) y= 2 cos(x) y= -4 sin(x)
Graphing
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13 What is the amplitude of y = 3cosx ?
Graphing
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14 What is the amplitude of y = 0.25cosx ?
Graphing
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15 What is the amplitude of y = -sinx ?
Graphing
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y= sin b(x) or y= cos b(x) In the study of transforming parent functions, we learned "b" was a horizontal stretch or shrink. y= cos x has b=1. Therefore cos x can make one complete cycle is 2#. For trig functions it is called the period.
Graphing
y = cos x completes 1 "cycle" in 2#. So the period is 2π. y = cos 2x completes 2 "cycles" in 2 # or 1 "cycle" in #. The period is # y = cos 0.5x completes
1/2 a cycle in 2#. The period is 4#.
Graphing
Graphing
16 What is the period of
A B C D
Graphing
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17 What is the period of
A B C D
Graphing
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18 What is the period of
A B C D
Graphing
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y= sin (x+c) or y= cos (x+c) In the study of transforming parent functions, we learned "c" was a horizontal shift y= cos (x+# ) has c = π. The graph of y= cos (x+π) is the graph of y=cos(x) shifted to the left # . For trig functions it is called the phase shift.
Graphing
y= sin (x) + d or y= cos (x) + d In the study of transforming parent functions, we learned "d" was a vertical shift
Graphing
23 What is the vertical shift in
Graphing
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24 What is the vertical shift in
Graphing
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25 What is the vertical shift in
Graphing
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30 What is the amplitude of this cosine graph?
Graphing
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31 What is the period of this cosine graph? (use 3.14 for pi)
Graphing
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32 What is the phase shift of this cosine graph?
Graphing
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33 What is the vertical shift of this cosine graph?
Graphing
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34 Which of the following of the following are equations for the graph?
A B C D
Graphing
Teacher Teacher
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When to use Law of Sines
(Recall triangle congruence statements) · ASA · AAS · SAS (use Law of Cosines) · SSS (use Law of Cosines) · SSA (use Law of Sines- but be cautious!)
Law of Sines
Teacher Teacher
Example: Teddy is driving toward the Old Man of the Mountain, the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain? 30 10 5280 x y
Law of Sines
Law of Sines with SSA. SSA information will lead to 0, 1,or 2 possible solutions. The one solution answer comes from when the bigger given side is
The 2 solution and no solution come from when sin
problem and the answer and its supplement are evaluated, sometimes both will work, sometimes one will work,and sometimes neither will work.
Teacher Teacher
Example solve triangle ABC A B C 40 5 7
Law of Sines
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Example solve triangle ABC A B C 40 7 5
Law of Sines
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A B C 40 7 5 64.1 Solution 1 115.9 A B C 40 7 5 Solution 2
Law of Sines
Teacher Teacher
Example solve triangle ABC A B C 50 14 7
Law of Sines
38 How many triangles meet the following conditions?
Law of Sines
Teacher Teacher
39 How many triangles meet the following conditions?
Law of Sines
Teacher Teacher
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When to use Law of Sines
(Recall triangle congruence statements) · ASA · AAS · SAS (use Law of Cosines) · SSS (use Law of Cosines) · SSA (use Law of Sines- but be cautious!) When we began to study Law of Sines, we looked at this table: Its now time to look at SAS and SSS triangles.
Law of Cosines
Example: Joe went camping. Sitting at his camp site he noticed it was 3 miles to one end of the lake and 4 miles to the other end. He determined that the angle between these two line of sites is 105
3 4 105 x
Law of Cosines
Teacher Teacher
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Trigonometry Identities are useful for simplifying expressions and proving
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Pythagorean Identities
Pythagorean Identities
Simplify:
Pythagorean Identities
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Simplify:
Pythagorean Identities
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Simplify:
Pythagorean Identities
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Prove:
Pythagorean Identities
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Prove:
Pythagorean Identities
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43 The following expression can be simplified to which choice?
A B C D
Pythagorean Identities
Teacher Teacher
44 The following expression can be simplified to which choice?
A B C D
Pythagorean Identities
Teacher Teacher
45 The following expression can be simplified to which choice?
A B C D
Pythagorean Identities
Teacher Teacher
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Angle Sum/Difference Identities are used to convert angles we aren't familiar with to ones we are (ie. multiples of 30, 45, 60, & 90).
Angle Sum/Difference
Sum/ Difference Identities
Angle Sum/Difference
Find the exact value of
Angle Sum/Difference
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Find the exact value of
Angle Sum/Difference
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Find the exact value of
Angle Sum/Difference
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Prove:
Angle Sum/Difference
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Prove:
Angle Sum/Difference
Teacher Teacher
46 Which choice is another way to write the given expression?
A B C D
Angle Sum/Difference
Teacher Teacher
47 Which choice is the exact value of the given expression?
A B C D
Angle Sum/Difference
Teacher Teacher
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Double-Angle Identities
Double Angle
Write cos3x in terms of cosx
Double Angle
Teacher Teacher
48 Which of the following choices is equivalent to the given expression?
A B C D
Double Angle
Teacher Teacher
50 Which of the following choices is equivalent to the given expression?
A B C D
Double Angle
Teacher Teacher
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Find the exact value of cos15 using Half-Angle Identity
Half Angle
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Find the exact value of tan 22.5
Half Angle
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51 Find the exact value of
A B C D
Half Angle
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52 Find the exact value of
A B C D
Half Angle
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Find cos(u/2) if sin u= - 3/7 and u is in the third quadrant
Pythagorean Identity but Why Negative?
Half Angle
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54 Find if and u is in the 4th quadrant?
A B C D
Half Angle
Teacher Teacher
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Power Reducing Identities
Power Reducing Identities
Reduce sin4x to an expression in terms of first power cosines.
Power Reducing Identities
Teacher Teacher
Reduce cos4x to an expression in terms of first power cosines.
Power Reducing Identities
Teacher Teacher
55 Which of the following choices is equivalent to the given expression?
A B C D
Power Reducing Identities
Teacher Teacher
56 Which of the following choices is equivalent to the given expression?
A B C D
Power Reducing Identities
Teacher Teacher
57 Which of the following choices is equivalent to the given expression?
A B C D
Power Reducing Identities
Teacher Teacher
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Sum to Product
Sum to Product
Write cos 11x + cos 9x as a product
Sum to Product
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Write sin 8x - sin 4x as a product
Sum to Product
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Find the exact value of cos
5π/ 12 + cos π/12
Sum to Product
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Prove
Sum to Product
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Prove:
Sum to Product
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58 Which of the following is equivalent to the given expression?
A B C D
Sum to Product
Teacher Teacher
59 Which of the following is equivalent to the given expression?
A B C D
Sum to Product
Teacher Teacher
60 Which of the following is not equivalent to the given expression?
A B C D
Sum to Product
Teacher Teacher
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Rewrite as a sum of trig functions.
Product to Sum
Teacher Teacher
Rewrite as a sum of trig functions.
Product to Sum
Teacher Teacher
61 Which choice is equivalent to the expression given?
A B C D
Product to Sum
Teacher Teacher
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Inverse Trig Functions
Since the cosine function does not pass the horizontal line test, we need to restrict its domain so that cos-1 is a function. cos x: Domain[0 , # ] Range[-1 , 1] cos-1 x: Domain[-1 , 1] Range[0 , π] Remember to find an inverse, switch x and y.
Inverse Trig Functions
Inverse Trig Functions
Inverse Trig Functions
Since the sine function does not pass the horizontal line test, we need to restrict its domain so that sin-1 is a function. sin x: Domain Range[-1 , 1] sin-1 x: Domain[-1 , 1] Range
Inverse Trig Functions
1
Inverse Trig Functions
Inverse Trig Functions
Since the tangent function does not pass the horizontal line test, we need to restrict its domain so that tan-1 is a function. tan x: Domain Range tan-1 x: Domain Range
Inverse Trig Functions
Inverse Trig Functions
Secant
Inverse Trig Functions
y=sec -1 x 1
sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: [0, # /2) ∪ [# , 3# /2)
Inverse Trig Functions
Cosecant
Inverse Trig Functions
1
Cosecant sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: (0, # /2] ∪ (# , 3# /2]
Inverse Trig Functions
Cotangent
Inverse Trig Functions
Cotangent 1
cot-1 x: Domain: Reals Range: (0 , # )
Inverse Trig Functions
Restrictions
Inverse Trig Functions
Example: Evaluate the following expression.
Inverse Trig Functions
Teacher Teacher
Example: Evaluate the following expression.
Inverse Trig Functions
Teacher Teacher
Example: Evaluate the following expressions.
Inverse Trig Functions
Teacher Teacher
63 Evaluate the following expression:
A B C D
Inverse Trig Functions
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64 Evaluate the following expression:
A B C D
Inverse Trig Functions
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65 Evaluate the following expression:
A B C D
Inverse Trig Functions
Teacher Teacher
Example: Evaluate the following expressions.
Inverse Trig Functions
Teacher Teacher
Example: Evaluate the following expressions.
Inverse Trig Functions
Teacher Teacher
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To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).
Examples: Solve.
Trig Equations
Teacher Teacher
To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
69 Find an apporoximate value of x on [0, ) that satisfies the following equation:
Trig Equations
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Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
Examples: Solve.
Trig Equations
Teacher Teacher
71 Find an apporoximate value of x on [0, ) that satisfies the following equation:
Trig Equations
Teacher Teacher