Post on 12-Jun-2020
Slide 1 / 162
Algebra II
Trigonometric Functions
www.njctl.org
2015-12-17
Slide 2 / 162
Trig Functions
· Graphing
· Radians & Degrees & Co-terminal angles
· Unit Circle
click on the topic to go to that section
· Trigonometric Identities
· Arc Length & Area of a Sector
Slide 3 / 162
Radians & Degrees
and Co-Terminal Angles
Return to Table of Contents
Slide 4 / 162
A few definitions:
A central angle of a circle is an angle whose vertex is the center of the circle.
An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle.
centralangle
interceptedarc
Slide 5 / 162
Radians and Degrees
One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are , or a little more than 6, radians in a circle.
Click on the circle for an animated view of radians.
Slide 6 / 162
Converting from Degrees to Radians
360∘ = 2 radians
1∘ = = 2360 180 radians
Example: Convert 50∘ and 90∘ to radians.
50∘ ⋅ 180 = 18
radians
90∘ ⋅ 180 = 2
radians
5
There are 360 in a circle. Therefore
Use this conversion factor to covert degrees to radians.
5
radians 2=
180 90∘
radians 18=
180 50∘
Example: Convert 50∘ and 90∘ to radians.
Slide 7 / 162
Converting from Radians to Degrees
2 radians = 360∘
1 radian = = 2
180
Example: Convert and to radians
360 degrees
4
4 180 ⋅ = 45∘
⋅ 180 = 180∘
Use this conversion factor to covert radians to degrees.
Slide 8 / 162
Converting between Radians and Degrees
Convert degrees to radians
Slide 9 / 162
Converting between Radians and Degrees
Convert radians to degrees
radians
radians
radians
Slide 10 / 162
Slide 11 / 162
Slide 12 / 162
Slide 13 / 162
4 Convert radians to degrees:
Slide 14 / 162
Angles
Initial side
Terminal side
Initial side
Terminal side
AngleAngle in standard position
An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side.
When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.
Slide 15 / 162
Positive Angle - terminal side rotates in a counter-clockwise direction
Negative Angle - terminal side rotates in a clockwise direction
α = - 37∘
Slide 16 / 162
Drawing angles in standard position
310∘
40∘
Each quadrant is 90∘, and 310∘ is 40∘ more than 270∘, so the terminal side is 40∘ past the negative y-axis.
500∘
500∘ is 140∘ more than 360∘, so the angle makes a complete revolution counterclockwise and then another 140∘.
Slide 17 / 162
Coterminal Angles
Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360∘ for degrees and 2 for radians.
Example: Find one positive and one negative angle that are terminal with 75∘.
75∘-285∘
435∘
75 + 360 = 435
75 - 360 = -285
Slide 18 / 162
5 Which angles are coterminal with 40∘? (Select all that are correct.)
A 320B -320C 400D -400
Slide 19 / 162
6 Which graph represents 425∘ ?
AB
C D
Slide 20 / 162
7 Which graph represents ?
A B
CD
Slide 21 / 162
8 Which angle is NOT coterminal with -55∘?
A 305∘
B 665∘
C -415∘
D -305∘
Slide 22 / 162
9 Which angle is coterminal with ?
A
B
C
D
Slide 23 / 162
Arc Length & Area of a Sector
Return to Table of Contents
Slide 24 / 162
Arc length: s = rArea of sector: A =
Arc length and the area of a sector
rarc length s
sector
How do these formulas relate to the area and the circumference of a circle?
(Measured in radians)
Area of sector: A = Arc length: s = r
Slide 25 / 162
40∘45∘
Who is getting more pie? Who is getting more of the crust at the outer edge?
Emily's slice is cut from a 9 inch pie.
Chester's slice is cut from an 8 inch pie.
(Assume both pies are the same height.)
(Try to work this out in your groups. The solution is on the next slide)
Slide 26 / 162
40∘45∘
The top of Emily's piece has an area of
The top of Chester's piece has an area of
Emily's crust has a length of
Chester's crust has a length of
click click
Slide 27 / 162
10 What is the top surface area of this slice of pizza from an 18-inch pie?
45∘
Slide 28 / 162
11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie?
45∘
Slide 29 / 162
12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth?
Slide 30 / 162
13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region?
8 inches
4 in
Slide 31 / 162
Unit CircleReturn toTable of
Contents
Slide 32 / 162
The circle x2 + y2 = 1, with center (0,0) and radius 1, is called the unit circle.
1
(1,0)
(0,-1)
(0,1)
(-1,0)
Quadrant I: x and y are both positive
Quadrant IV: x is positive and y is negative
Quadrant III: x and y are both negative
Quadrant II: x is negative and y is positive
The Unit Circle
Slide 33 / 162
In this triangle,
sin#= = b
θa
b
(a,b)
(1,0)
(0,-1)
(-1,0)
(0,1)
b1
cos# = = aa1
so the coordinates of (a,b) are also
(cos#, sin#)
1
The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures.
For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point.
Slide 34 / 162
In this example, the terminal point is in Quadrant IV.
For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ).
0.57
0.82-55
1
If we look at the triangle, we can see that
sin(-55 ) = 0.82
cos(-55 ) = 0.57
EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis.
Slide 35 / 162
Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:
Slide 36 / 162
1
In this example, we know the angle. Using a calculator, we find that cos 44∘ ≈ .72 and sin 44∘ ≈ .69, so the coordinates of C are approximately (0.72, 0.69).
Note that 0.722 + 0.692 ≈ 1!
What are the coordinates of point C?
Slide 37 / 162
The Tangent Function
Recall SOH-CAH-TOA
hypotenuse
adjacent side
oppo
site
sid
e
#
sin = # opphyp
cos = # adjhyp
tan = # oppadj
It is also true that tan = . # sin cos ##
Why? opphypadjhyp
= ⋅ = = tan opphyp adj
hyp oppadj #
Slide 38 / 162
Angles in the Unit Circle
Example: Given a terminal point on the unit circle (- ).
Find the value of cos, sin and tan of the angle.
Solution: Let the angle be .
x = cos , so cos = .
y = sin , so sin = .
tan = = = = ⋅
(Shortcut: Just cross out the 41's in the complex fraction.)
Slide 39 / 162
Example: Given a terminal point , find #, tan# and csc#.
To find #, use sin-1 or cos -1 :
sin -1 ( ) = #
# # 28.1∘tan# = sin#/ cos#
tan # =
csc# = 1/ sin#
csc # = Note the "hidden" Pythagorean Triple, 8, 15, 17).
Slide 40 / 162
(__, - ) 513
A
θ22.3 -22.3
Example: Find the x-value of point A, θ and the tan θ. For every point on the circle,
Since x is in quadrant III, x = -1213
sin-1(- ) ≈ -22.3 , BUT θ is in quadrant III, so θ = 180 + 22.3 = 202.3 (notice how 202.3 and -22.3 have the same sine)
513
tan θ = = = sin θcos θ
512
Slide 41 / 162
Example: Given the terminal point of (-5/13,-12 /13). Find sin x, cos x, and tan x.
Slide 42 / 162
14 What is tan θ?
A
B
C
D
(- ,__)35
θ
Slide 43 / 162
15 What is sin θ?
A
B
C
D
(- ,__)35
θ
Slide 44 / 162
16 What is θ (give your answer to the nearest degree)?
(- ,__)35
θ
Slide 45 / 162
17 Given the terminal point , find tan x.
Slide 46 / 162
Slide 47 / 162
Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45 . A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values.
Isosceles Right Triangle Equilateral Triangle
(the altitude divides the triangle into two 30-60-90 triangles)
Slide 48 / 162
Special Right Triangles
(see Triangle Trig Review unit for more detail on this topic)
Slide 49 / 162
1
( , )
45∘
Special Triangles and the Unit Circle
(- , )
-
1
45∘
Multiples of 45 angles have sin and cos of ± ,
depending on the quadrant.
Slide 50 / 162
30o
45o
60o
30o
45o
60o
30o
45o
60o
30o
45o
60o
Slide 51 / 162
Drag the degree and radian angle measures to the angles of the circle:
#4
3#4
5#4
3#2
#2# 7#
42#0
0∘
45∘
90∘
135∘
180∘
225∘
270∘
315∘
360∘
Slide 52 / 162
#4
3#4
5#4
3#2
#2
#
7#4
2#0
( , )
( , )
( , )
( , )
( , )
( , )
( , )
( , )
0
1
-1
Fill in the coordinates of x and y for each point on the unit circle:
Slide 53 / 162
30∘
( , )
60∘
( , )
11
Special Triangles and the Unit Circle
Angles that are multiples of 30 have sin and cos of ± and ± .
Slide 54 / 162
Drag the degree and radian angle measures to the angles of the circle:
#6
2#3
5#3
3#2
#2
# 7#6
2#0 11#6
4#3
#3
5#60∘
30∘
60∘
90∘
120∘
150∘
180∘
210∘
240∘
270∘
300∘
330∘
360∘
Slide 55 / 162
#6
2#3
5#3
3#2
#2
#
7#6
2#0
11#6
4#3
#3
5#6
( , )
( , )
( , )
( , )( , )( , )
( , )
( , )
( , )
( , )
Drag in the coordinates of x and y for each point on the unit circle:
( , )
( , )
0
1
-1
Slide 56 / 162
Special Angles in Degrees
Slide 57 / 162
Radian Values of Special Angles
Slide 58 / 162
Exact Values of Special Angles
Slide 59 / 162
Put it all together...
Slide 60 / 162
Degrees 0∘ 30∘ 45∘ 60∘ 90∘
Radians
sin θ cos θ
tan θ
Complete the table below:
Exact values of special angles
Slide 61 / 162
Slide 62 / 162
Slide 63 / 162
If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values.
Slide 64 / 162
opp
adj
hyp
-3
Example: If tan = , and sin < 0, find sin , cos and the value of .
Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III.
· Draw a right triangle in Quadrant III.
· Use the Pythagorean Theorem to find the length of the hypotenuse:
(Continued on next slide)
Slide 65 / 162
opp
adj
hyp
-3Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values.
sin = =
cos = =
Use any inverse trig function to find the angle. tan-1( ) ≈ 36.7 . Because the angle is in QIII, we need to add 180 + 36.7 = 216.7, so ≈ 217 .
Slide 66 / 162
Slide 67 / 162
Slide 68 / 162
Slide 69 / 162
22 Which functions are 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
Slide 70 / 162
23 Which functions are 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
Slide 71 / 162
24 Which functions are 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
Slide 72 / 162
Graphing Trig
Functions Return toTable of
Contents
Slide 73 / 162
If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions:
(Once the webpage opens, click on Download)
Slide 74 / 162
Graphing the Sine Function, y = sin x
Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of , and the corresponding values of y or sin x.
(Remember, is just a bit more than 3.)
Since the values are based on a circle, values will repeat.
Slide 75 / 162
Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2 .
Slide 76 / 162
Graphing the Cosine Curve
Since the values are based on a circle, values will repeat.
Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2 .
Slide 77 / 162
Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2 .
Slide 78 / 162
Compare the graphs:
How are they similar and how are they different?
y = sin x y = cos x
Slide 79 / 162
Characteristics of y = sin x and y = cos x
Domain: set of real numbers (x can be anything)
Range: -1 ≤ y ≤ 1
Amplitude: one-half the distance from the minimum of the graph to the maximum or 1.
The functions are periodic - the pattern repeats every 2 units.
amplitude = 1
period = 2
range:-1 ≤ y ≤ 1
Slide 80 / 162
Predict, Explore, Confirm
1. Using your prior knowledge of transforming functions, predict what happens to the following functions:
2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.
3. Do your conclusions match your predictions?
Slide 81 / 162
y = a s in x or y = a cos x
Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, or the distance from the midline to the maximum.
Slide 82 / 162
Consider the graphs of y = s in x y = 2s in x y = s in x
y = s in x
y = 2s in x
y = s in x
What do you notice about these graphs? What does the value of "a" do to the graph?
Name the amplitude of each graph.
Slide 83 / 162
As shown in the graph below, the graph of y = -3cos x is a reflection over the x-axis of the graph of y = 3cos x.
What is the amplitude of each function?y = 3cos x y = -3cos x
The domain of each function is the set of real numbers and the range is {x|-3 ≤ x ≤ 3}.
Slide 84 / 162
Sketch each graph on the interval from 0 to 2 :
y = 4cos x y = -.25 sin x
Slide 85 / 162
25 What is the amplitude of y = 3cos x ?
Slide 86 / 162
26 What is the amplitude of y = 0.25cos x ?
Slide 87 / 162
27 What is the amplitude of y = -s in x ?
Slide 88 / 162
28 What is the range of the function y = 2s in x ?
A All real numbersB -2 < x < 2C 0 ≤ x ≤ 2D -2 ≤ x ≤ 2
Slide 89 / 162
29 What is the domain of y = -3cos x ?
A All real numbersB -3 < x < 3C 0 ≤ x ≤ 3D -3 ≤ x ≤ 3
Slide 90 / 162
30 Which graph represents the function y = -2s in x?
AB
C D
Slide 91 / 162
31 What is the amplitude of the graph below?
Slide 92 / 162
Predict, Explore, Confirm
1. Using your prior knowledge of transforming functions, predict what happens to the following functions:
2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.
3. Do your conclusions match your predictions?
Slide 93 / 162
A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle.
The trig functions are periodic functions.
The basic sine and cosine curves have a period of 2 , meaning that the graph completes one complete cycle in 2 units.
Slide 94 / 162
y = s in bx or y = cos bx
Consider the graphs of y = cos x and y = cos 2x.
y = cos x
y = cos 2xone cycle
Notice that the graph of y = cos 2x completes one cycle twice as fast, or in units.
Slide 95 / 162
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 a cycle in 4# . The period is 4# .
Slide 96 / 162
The period for y = cos bx or y = s in bx is
2 b
y = cos x b = 1 P = = 2
y = cos 2x b = 2 P = =
y = cos 0.5x b = 0.5 P = = 4
P =
2 1
2
2
2
0.5
2
b2
Slide 97 / 162
32 What is the period of
A
B
C
D
Slide 98 / 162
33 What is the period of
A
B
C
D
Slide 99 / 162
34 What is the period of
A
B
C
D
Slide 100 / 162
Sketch the graph of each function from x = 0 to x = 2 .
y = 2cos 3x
y = -2cos 2xy = s in 2x
y = cos x
Slide 101 / 162
35 What is the period of the graph below?
A
B 2
C 3
D
2
2
Slide 102 / 162
36 What is the period of the graph shown?
A
B
C
D
2
3
23
Slide 103 / 162
37 What is the equation of this function?
AB
C
D
y = sin 3xy = cos 3xy = 3cos xy = 3sin x
Slide 104 / 162
Predict, Explore, Confirm
1. Using your prior knowledge of transforming functions, predict what happens to the following functions:
2. Using your graphing calculator, insert the parent function into and the transformed function into . Compare the graphs.
3. Do your conclusions match your predictions?
Slide 105 / 162
Translating Sine and Cosine Functions
Trig functions can be translated in the same way as any other function.
The horizontal shift is called a phase shift.
What are your conclusions from the graphing calculator activity?
Slide 106 / 162
k
Horizontal or phase shift
Vertical shift
Drag each equation to the matching graph
y = cos x
y = cos (x + )2
y = sin x + 2y = sin x
Slide 107 / 162
Slide 108 / 162
Slide 109 / 162
Consider the graphs of
In order to determine the phase shift, the coefficient of x must be factored out.
In , the 2 is factored out. The phase shift is .
In , when the 2 is factored out, we get . The phase shift is .
and (which is which?)
Slide 110 / 162
Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable.
Example: Set
Solve for x:
So, the phase shift is 2 .
Slide 111 / 162
Slide 112 / 162
Slide 113 / 162
y= sin (x) + k or y= cos (x) + kVertical ShiftThe k moves the graph up or down.
The graph below is of the equation y = 2 sin (3x).
The midline of this graph is the horizontal line y = 0.
Sketch the graph of y = 2 sin (3x) + 1.
Slide 114 / 162
42 What is the vertical shift in
Slide 115 / 162
43 What is the vertical shift in
Slide 116 / 162
44 What is the vertical shift in
Slide 117 / 162
Slide 118 / 162
Graphing a Sine or Cosine Function:
Step 1: Identify the amplitude, period, phase shift and vertical shift.
Step 2: Draw the midline (y = k)
Step 3: Find 5 key points - maximums, minimums and points on the midline
Step 4: Draw the graph through the 5 points.
Slide 119 / 162
Example:
Step 1: Amplitude: |-1| = 1
Period:
Phase Shift:
Vertical Shift: 2 (up 2)
Slide 120 / 162
Step 2: Draw the midline y = 2Step 3: Find the 5 key points
Note: for x, adding comes from dividing the cycle, 3 by 4.
For y, adding and subtracting 1 comes from the amplitude.
Slide 121 / 162
Step 4: Graph
Slide 122 / 162
You try:
Slide 123 / 162
Slide 124 / 162
Slide 125 / 162
Slide 126 / 162
Slide 127 / 162
49 What is the amplitude of this cosine graph?
Slide 128 / 162
50 What is the period of this cosine graph? (use 3.14 for pi)
Slide 129 / 162
51 What is the vertical shift of this cosine graph?
Slide 130 / 162
52 Which of the following of the following is an equation for the graph?
A
B
C
D
Slide 131 / 162
The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1 - 12. Sketch the graph of this equation. What is the average temperature in June?
Slide 132 / 162
Graphing the Tangent FunctionGraph by creating a table of values of key points. One option is to use the set of values for x that are multiples of , and the corresponding values of y or tan x.
Slide 133 / 162
Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction.
This is shown on the graph by the vertical lines, or asymptotes at these x values.
The period of the function is units, because there is one complete cycle from to . As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.
This is shown on the graph by the vertical lines, or asymptotes at these x values.
The period of the function is units, because there is one complete cycle from to . As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.
Slide 134 / 162
Example: Sketch the graph of y = tan (x + ) + 2
Asymptotes will be at 0, , 2 , etc. The midline will be at y = 2.
Slide 135 / 162
53 Which graph represents y = -tan x?
A B
C D
Slide 136 / 162
Trigonometric Identities
Return toTable of
Contents
Slide 137 / 162
Key Ideas
An identity is a mathematical equation that is true for all defined values of the variable.
A trigonometric identity is an identity that contains one or more trig ratios.
By contrast, a conditional equation is one that is only true for a limited set of values.
By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities.
Slide 138 / 162
Drag each equation into the correct box:
Identities Conditional Equations
3x + 4 = 93x + 4 = 3x + 4
sin # + cos # = 1
2x5 =x3
tan θ cot θ =1 2(x-1) = 2x - 2
(x + 3)2 = x2 + 9 sin 4x = 4sin x
5x - 7y = -(7y - 5x)
Slide 139 / 162
Basic Trig Identities
csc # = 1 sin #
sec # = 1 cos #
cot # = 1 tan #
sin # = 1 csc # 1 1 cos # = 1
sec #tan # = 1
cot #
Reciprocal Identities
Tangent Identity
Cotangent Identity
tan # = sin # cos #
cot # = cos # sin #
Slide 140 / 162
By using the basic identities we can change an expression into an equivalent expression.
Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.
Slide 141 / 162
Algebraic example Trig example
(x - y)(x + y) = x 2 - y2 (1 - cos #)(1 + cos #) = 1 - cos2#
Slide 142 / 162
Recall the unit circle, x2 + y2 = 1.
1
(1,0)
(0,-1)
(0,1)
(-1,0)
(cos #,sin #)
Pythagorean Identities
For any point (x, y) on the circle, its coordinates are
(cos #, sin #).
Therefore,
(cos #)2 + (sin #)2 = 12, which is usually written as
cos2 θ + sin2 θ = 1
Slide 143 / 162
Pythagorean Identities
How do we transform the first identity, which is derived from the unit circle, to the other two?
Slide 144 / 162
Alternative Forms of Identities
Since we know that 3 + 5 = 8, we also know that
8 - 5 = 3 and 8 - 3 = 5. In elementary school we call these equivalent equations "fact families".
Similarly, if
cos2 θ + sin2 θ = 1, it follows that
1 - cos2 θ = sin2 θ and
1 - sin2 θ = cos2 θ.
Slide 145 / 162
More Alternative Forms
Another fact family tells that since = 4, it follows that 4 ⋅ 5 = 20.
Since sec θ = , then sec θ cos θ = 1 (multiply both sides of the first equation by cos #).
205
1 cos θ
Slide 146 / 162
Simplifying Trig Expressions
Example 1: Simplify csc θ cos θ tan θ.
Rewrite each trig ratio in terms of cos and sin:
1 sin #sin θ
⋅ cos θ ⋅ = 1cos #
(When multiplying fractions, it is often easier to reduce or cancel before you multiply.)
Example 2: Simplify csc2 θ(1 - cos2 θ). 1 sin2 θ ⋅ (sin2 θ) = 1
Slide 147 / 162
Verifying an Identity
Transform one side of the identity to be the same as the other side
sin # cot # = cos # sin # ⋅ = cos # cos #
sin #
Example 1: Verify
Example 2: Verify
cos θ csc θ tan θ = 1
cos # ⋅ ⋅ = 1 1 sin #
sin # cos #
Slide 148 / 162
Simplify:
Slide 149 / 162
Simplify:
Slide 150 / 162
Simplify:
Slide 151 / 162
Slide 152 / 162
Simplify:
Slide 153 / 162
Verify:
Slide 154 / 162
Slide 155 / 162
Verify:
Slide 156 / 162
Slide 157 / 162
Slide 158 / 162
54 Which equation is NOT an identity?
A sin2 x= 1 - cos2 xB 2 cot x = C tan2 x = sec2 x - 1D sin2 x = cos2 x - 1
2cos xsin x
Slide 159 / 162
55 The following expression can be simplified to which choice?
A
B
C
D
Slide 160 / 162
56 The following expression can be simplified to which choice?
A
B
C
D
Slide 161 / 162
57 The following expression can be simplified to which choice?
A
B
C
D
Slide 162 / 162