Slide 2 / 207 Pre-Calc -...
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Pre-Calc
Trigonometry
www.njctl.org
2015-03-24
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Table of Contents
Unit CircleGraphingLaw of SinesLaw of CosinesPythagorean IdentitiesAngle Sum/DifferenceDouble AngleHalf Angle
Sum to Product
Inverse Trig FunctionsTrig Equations
Product to Sum
Power Reducing
click on the topic to go to that section
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Unit Circle
Return toTable ofContents
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Unit Circle
Goals and ObjectivesStudents will understand how to use the Unit Circle to find angles and determine their trigonometric value.
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Why do we need this?The Unit Circle is a tool that allows us to determine the location of any angle.
Unit Circle
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Special Right Triangles
Unit Circle
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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
hg
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30o
45o
60o
30o
45o
60o
30o
45o
60o
30o
45o
60o
Unit Circle
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Unit Circle
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Unit Circle
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Unit Circle
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Unit Circle
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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
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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
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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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Example: Given the terminal point of (-5/13,-12 /13) find sin x, cos x, and tan x.
Unit Circle
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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 2nd quadrant
Unit Circle
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Graphing
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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
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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
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Graphing cos, sin, & tanGraph by using values from the table.Since the values are based on a circle, values will repeat.
Graphing
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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
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Parts of a trig graph
x
cos x Amplitude
Period
Graphing
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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
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In y= cos(x), a=1This 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
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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
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The period for y= cos bx or y= sin bx is
Graphing
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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
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18 What is the period of
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D
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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
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y= sin (x) + d or y= cos (x) + d
In the study of transforming parent functions, we learned "d" was a vertical shift
Graphing
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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?
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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
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Law of Sines
Return toTable ofContents
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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
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the angle of elevation is 10 degrees, he drives another mile and the angle of elevation is 30 degrees. How tall is the mountain?
30105280
xy
Law of Sines
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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 opposite the given angle.
The 2 solution and no solution come from when sin-1 is used in the problem and the answer and its supplement are evaluated, sometimes both will work, sometimes one will work,and sometimes neither will work.
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A
B
C405 7
Law of Sines
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Example solve triangle ABC
A
B
C40
7 5
Law of Sines
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A
B
C40
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64.1
Solution 1
115.9A
B
C407 5
Solution 2
Law of Sines
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Example solve triangle ABC
A
B
C50
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Law of Sines
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38 How many triangles meet the following conditions?
Law of Sines
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39 How many triangles meet the following conditions?
Law of Sines
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Law of Cosines
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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
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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 degrees. How far is it across the lake?
3 4105
x
Law of Cosines
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Identities
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Trigonometry Identities are useful for simplifying expressions and proving
other identities.
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Pythagorean Identities
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Trigonometric Ratios
Pythagorean Identities
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Pythagorean Identities
Pythagorean Identities
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Simplify:
Pythagorean Identities
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Simplify:
Pythagorean IdentitiesTe
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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
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44 The following expression can be simplified to which choice?
A
B
C
D
Pythagorean Identities
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45 The following expression can be simplified to which choice?
A
B
C
D
Pythagorean Identities
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Angle Sum/Difference
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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
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Sum/ Difference Identities
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
Teac
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Find the exact value of
Angle Sum/DifferenceTe
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rTe
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r
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Prove:
Angle Sum/Difference
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Prove:
Angle Sum/Difference
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46 Which choice is another way to write the given expression?
A
B
C
D
Angle Sum/DifferenceTe
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rTe
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47 Which choice is the exact value of the given expression?
A
B
C
D
Angle Sum/Difference
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Double Angle
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Double-Angle Identities
Double Angle
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Write cos3x in terms of cosx
Double Angle
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48 Which of the following choices is equivalent to the given expression?
A
B
C
D
Double Angle
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50 Which of the following choices is equivalent to the given expression?
A
B
C
D
Double Angle
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Half Angle
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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 butWhy Negative?
Half Angle
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54 Find if and u is in the 4th quadrant?
A
B
C
D
Half Angle
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Power Reducing Identities
Return toTable ofContents
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Power Reducing Identities
Power Reducing Identities
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Reduce sin4x to an expression in terms of first power cosines.
Power Reducing Identities
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Reduce cos4x to an expression in terms of first power cosines.
Power Reducing IdentitiesTe
ache
rTe
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55 Which of the following choices is equivalent to the given expression?
A
B
C
D
Power Reducing Identities
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56 Which of the following choices is equivalent to the given expression?
A
B
C
D
Power Reducing Identities
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57 Which of the following choices is equivalent to the given expression?
A
B
C
D
Power Reducing Identities
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Sum to Product
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Sum to Product
Sum to Product
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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
Teac
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Prove
Sum to Product
Teac
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Teac
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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
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59 Which of the following is equivalent to the given expression?
A
B
C
D
Sum to Product
Teac
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Teac
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60 Which of the following is not equivalent to the given expression?
A
B
C
D
Sum to Product
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Product to Sum
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Slide 161 / 207
Rewrite as a sum of trig functions.
Product to Sum
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Rewrite as a sum of trig functions.
Product to Sum
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61 Which choice is equivalent to the expression given?
A
B
C
D
Product to SumTe
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rTe
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Inverse Trig Functions
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Slide 167 / 207
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Inverse Trig FunctionsSince 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
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1-1
#
# /2
y=cos-1x
Inverse Trig Functions
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Inverse Trig FunctionsSince 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
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y=sin-1x
1-1
Inverse Trig Functions
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Inverse Trig FunctionsSince 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
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y=tan-1x
Inverse Trig Functions
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Secant
Inverse Trig Functions
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y=sec -1 x
1-1
sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: [0, # /2) ∪ [# , 3# /2)
Inverse Trig Functions
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Cosecant
Inverse Trig Functions
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1-1
Cosecant
sec-1x : Domain: (-# ,-1] ∪ [1 , # ) Range: (0, # /2] ∪ (# , 3# /2]
Inverse Trig Functions
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Cotangent
Inverse Trig Functions
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Cotangent
1-1cot-1 x: Domain: Reals Range: (0 , # )
Inverse Trig Functions
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Restrictions
Inverse Trig Functions
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Example: Evaluate the following expression.
Inverse Trig FunctionsTe
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rTe
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Example: Evaluate the following expression.
Inverse Trig Functions
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Example: Evaluate the following expressions.
Inverse Trig Functions
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63 Evaluate the following expression:
A
B
C
D
Inverse Trig Functions
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Teac
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64 Evaluate the following expression:
A
B
C
D
Inverse Trig Functions
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Teac
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65 Evaluate the following expression:
A
B
CD
Inverse Trig Functions
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Teac
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Example: Evaluate the following expressions.
Inverse Trig Functions
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Teac
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Example: Evaluate the following expressions.
Inverse Trig Functions
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Teac
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Slide 191 / 207
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Trig Equations
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To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).
Examples: Solve.
Trig EquationsTe
ache
rTe
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To solve a trigonometry equation, apply the rules of algebra to isolate the trig function(s).
Examples: Solve.
Trig Equations
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Examples: Solve.
Trig Equations
Teac
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Teac
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Examples: Solve.
Trig EquationsTe
ache
rTe
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69 Find an apporoximate value of x on [0, ) that satisfies the following equation:
Trig Equations
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Teac
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Examples: Solve.
Trig EquationsTe
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rTe
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r
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Examples: Solve.
Trig Equations
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Teac
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Examples: Solve.
Trig Equations
Teac
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Teac
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Examples: Solve.
Trig Equations
Teac
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Teac
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Examples: Solve.
Trig Equations
Teac
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Teac
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Examples: Solve.
Trig Equations
Teac
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Teac
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71 Find an apporoximate value of x on [0, ) that satisfies the following equation:
Trig Equations
Teac
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Teac
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