AP Calculus Summer Packet
Name (Print) _______________________________
Year End Grade for Honors Pre-Calculus (Number)_____________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find an equation for the vertical line and the horizontal line through the given point.
1) 5, - 1
7
A) x = 5
y=- 1
7
2) (
3) (0,
7, 1.3)
A) x = - 7
y = -1.3
3)
A) x = 0
y=
B) x = - 5
y=- 1
7
C) x = 5
y =-7
D) x = - 1
7
B) x = 1.3
D) x = -1.3
7
C) x =
7
y = 1.3
3
C) x = y =0
D) x =
3
y =0
y=
B) x = 0
3
4) (-π, 0)
A) x = π
y =0
y=-
B) x = 0
y= π
y=5
3
y=-
7
C) x = -π
y =0
D) x = 0
y = -π
C) y = -x - 2
D) y = x + 2
C) y = - 2 x + 1
5
7
D) y = - 2 x - 7
5
7) Passes through (-6, -5) and has slope 0
A) x = -6
B) y = -5
C) x = -5
D) y = -6
8) Passes through (-3, -6) and has no slope
A) y = -6
B) x = -6
C) x = -3
D) y = -3
Write an equation for the line described.
5) Passes through (5, 3) with slope 1
A) y = x + 8
B) y = x - 2
6) Passes through (5, 5) with slope - 2
5
A) y = - 2 x + 7
5
B) y = - 5 x - 1
2
7
1
9) Passes through (7, -3) and (-1, 6)
A) y = 9 x + 39
B) y = - 9 x + 39
8
8
8
8
10) Passes through (-5, 0) and (6, 8)
A) y = - 5 x - 7
B) y = 5 x - 7
2
2
C) y = 10 x + 32
7
7
D) y = - 10 x + 32
7
7
C) y = - 8 x + 40
11
11
D) y = 8 x + 40
11
11
C) y = 5
D) y = 1 x + 5
2
C) y = 1 x + 3
3
D) y = 1 x - 3
3
11) Has slope 1 and y-intercept 5
2
A) y = 2x + 5
B) y = 1 x - 5
2
12) Has y-intercept -3 and x-intercept 1
A) y = 3x - 3
B) y = 3x + 3
13) Passes through (-5, -3) and is parallel to the line 8x + 7y = -19
A) y = 5 x - 19
B) y = - 8 x - 61
C) y = 8 x + 61
7
7
7
7
7
7
D) y = - 7 x - 3
8
8
14) Passes through (5, -1) and perpendicular to the line -7x + 8y = -43
A) y = - 8 x + 33
B) y = - 8 x
C) y = - 7 x + 33
7
7
7
8
D) y = 8 x + 33
7
7
Find the slope and the y-intercept of the line.
15) 4x - 5y = -18
A) m = - 5 ; y-intercept: (0, -5)
4
B) m = 4 ; y-intercept: 0, 18
5
5
C) m = - 4 ; y-intercept: (0, -18)
5
D) m = 5 ; y-intercept: 0, 18
4
5
16) x + 2y = 3
A) m = 1 ; y-intercept: (0, 6)
2
B) m = 2; y-intercept: (0, 6)
C) m = - 1 ; y-intercept: 0, 3
2
2
D) m = -2; y-intercept: (0, 0)
17) 3x - 5y = -25
A) m = 0; y-intercept: (0, 3)
B) m = 3 ; y-intercept: (0,5)
5
C) m = 3 ; y-intercept: (0, -25)
5
D) m = -3; y-intercept: (0, -5)
2
18) 4y + 2x = 4
A) m = - 1 ; y-intercept: 0, 1
2
B) m = 1 ; y-intercept: (0, 0)
2
C) m = 4; y-intercept: (0, 0)
D) m = -2; y-intercept: (0, 4)
Solve the problem.
19) Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they
entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on
entering the program versus their current GPAs. Find the linear regression equation for the data.
Entering GPA (x)
3.5
3.8
3.6
3.6
3.5
3.9
4.0
3.9
3.5
3.7
A) y = 0.497x + 5.81
Current GPA (y)
3.6
3.7
3.9
3.6
3.9
3.8
3.7
3.9
3.8
4.0
B) y = 0.0313x + 3.67
C) y = 0.0212x + 4.91
D) y = 0.329x + 2.51
20) A study was conducted to compare the average time spent in the lab each week versus course grade for
computer students. The results are recorded in the table below. Find the linear regression equation for the
data.
Number of hours spent in lab (x)
10
11
16
9
7
15
16
10
A) y = - 1.86x + 88.6
Grade (percent)(y)
96
51
62
58
89
81
46
51
B) y = 44.3x + 0.930
C) y = 0.930x + 44.3
D) y = 88.6x + 1.86
21) Two separate tests are designed to measure a student's ability to solve problems. Several students are
randomly selected to take both tests and the results are shown below. Find the linear regression equation for
the data.
Test A (x) 48 52 58 44 43 43 40 51 59
Test B (y) 73 67 73 59 58 56 58 64 74
A) y = 0.930x + 19.4
C) y = - 0.930x - 19.4
B) y = - 19.4x + 0.930
D) y = 19.4x - 0.930
3
22) Find an equation for the least squares line representing weight, in pounds, as a function of height, in inches,
of men. Then, predict the weight of a man who is 68 inches tall to the nearest tenth of a pound. The
following data are the (height, weight) pairs for 8 men: (66, 150), (68, 160), (69, 166), (70, 175), (71, 181), (72,
191), (73, 198), (74, 206).
A) 165.1 pounds
B) 151.4 pounds
C) 160.0 pounds
D) 161.2 pounds
Find the formula for the function.
23) Express the perimeter of a square as a function of the square's side length x.
A) P(x) = 6x
B) P(x) = 3x
C) P(x) = 4x
2
D) P(x) = x3
24) Express the area of a square as a function of its side length x.
A) A(x) = 4x
B) A(x) = x2
C) A(x) = x4
D) A(x) = 2x
25) Express the length d of a square's diagonal as a function of its side length x.
A) d(x) = x 3
B) d(x) = x
C) d(x) = 2x
D) d(x) = x
2
26) Express the perimeter of an isosceles triangle with side lengths x, 5x, and 5x as a function of the side length.
A) P(x) = 11x
B) P(x) = 10x
C) P(x) = 25x3
D) P(x) = 10x3
27) Express the area of a circle as a function of its radius r.
A) A(r) = πr
B) A(r) = 2πr
C) A(r) = πr2
28) Express the volume of a sphere as a function of its radius r.
A) V(r) = 4 πr3
B) V(r) = 2 πr2
C) V(r) = 3 πr3
3
3
4
Find the domain and range.
29) y = x2 - 10
A) Domain: [0, ∞), Range: (-∞, -10]
C) Domain: (-∞, ∞), Range: [-10,∞)
D) V(r) = πr3
B) Domain: (-∞, ∞), Range: (-∞, ∞)
D) Domain: [-100,∞), Range: [-10,∞)
30) y = 3 - x
A) Domain: [0,∞), Range: (-∞,3]
C) Domain: (-∞,3], Range: (-∞, ∞)
31) y =
D) A(r) = πr3
B) Domain: (-∞, ∞), Range: (-∞,3]
D) Domain: (-∞,0], Range: [3,∞)
-1
x+1
A) Domain: [1,∞), Range: (-∞,∞)
C) Domain: (-1,∞), Range: (-∞,0)
B) Domain: [0,∞), Range: (-∞,∞)
D) Domain: (-∞,-1), Range: (0,∞)
32) y = 8 + x
A) Domain: (-∞, ∞); Range: y ≥ [-8, ∞)
C) Domain: (-∞, ∞); Range: (-∞, ∞)
B) Domain: [-8, ∞); Range: [0, ∞)
D) Domain: : [0, ∞); Range: (-∞, ∞)
4
33) f(x) =
14
14 - x
A) Domain: (-∞,
B) Domain: (-∞,
C) Domain: (-∞,
D) Domain: (-∞,
14) ∪ (14, ∞); Range: (-∞, ∞)
∞); Range: (-∞, 0) ∪ (0, ∞)
∞); Range: (-∞, ∞)
14) ∪ (14, ∞); Range: (-∞, 0) ∪ (0, ∞)
Determine if the function is even, odd, or neither.
34) y = (x + 8)(x - 7)
A) Even
B) Odd
35) y =
-8
x2 - 9
A) Even
36) y =
B) Odd
C) Neither
B) Odd
C) Neither
B) Odd
C) Neither
B) Odd
C) Neither
7
x-1
A) Even
37) y =
C) Neither
5x
x2 - 9
A) Even
38) y = x2 - 3
A) Even
5
Graph the piecewise-defined function.
x≤2
39) f(x) = 4 - x,
1 - 2x,
x>2
A)
B)
y
-10 -8
-6
-4
y
10
10
8
8
6
6
4
4
2
2
-2
2
4
6
8
10 x
-10 -8
-6
-4
-2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
C)
2
4
6
8
10 x
2
4
6
8
10 x
D)
y
y
10
8
8
6
6
-10 -8
-6
-4
4
4
2
2
-2
2
4
6
8
10 x
-10 -8
-2
-6
-4
-2
-2
-4
-4
-6
-6
-8
-8
-10
6
Find a formula for the function graphed.
40)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
2,
-2,
x<0
x ≥0
B) f(x) =
2x,
-2x,
x≤0
x>0
C) f(x) =
-2,
2,
x≤0
x>0
D) f(x) =
2,
-2,
x≤0
x>0
B) f(x) =
4,
-x,
x<0
x ≥0
D) f(x) =
4,
x,
41)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
4,
-4x,
C) f(x) =
4,
-x,
x<0
x ≥0
x≤0
x>0
7
x<0
x ≥0
42)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
2,
4 - x,
x<0
x ≥0
B) f(x) =
2,
4 - x,
x≤1
x>1
C) f(x) =
2,
4 - x,
x<1
x>1
D) f(x) =
2,
x - 4,
x<1
x≥1
B) f(x) =
5 + x,
-3
5 - x,
-3
43)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
C) f(x) =
5 + x,
-3
5 - x,
-3
x≤2
x>2
x≤2
x>2
D) f(x) =
8
x<2
x>2
x<2
x ≥2
44)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
-2x,
x + 2,
x≤1
x>1
C) f(x) =
x,
2x + 1,
x≤1
x>1
B) f(x) =
-2x,
x + 1,
x≤1
x>1
D) f(x) =
2x,
x + 1,
x≤1
x>1
45)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A) f(x) =
x,
6 - x,
0≤x≤3
3<x≤6
B) f(x) =
6 - x,
x,
0≤x≤3
3<x≤6
C) f(x) =
x + 6,
-x,
0≤x≤3
3<x≤6
D) f(x) =
-x,
x + 6,
0≤x≤3
3<x≤6
Solve the problem.
46) If f(x) = 4x 2 + 2x + 7 and g(x) = 2x - 4, find g(f(x)).
A) 4x2 + 4x + 10
B) 8x2 + 4x + 10
C) 8x2 + 4x + 18
D) 4x2 + 2x + 3
47) If f(x) = 9x - 5 and g(x) = 3x2 + 2x - 6, find g(f(-3)).
A) -166
B) 130
C) -194
D) 3002
9
48) If f(x) = 9x + 5 and g(x) = 2x2 + 3x - 7, find g(f(2)).
A) 108
B) 68
C) 32
D) 1120
49) If (f∘g)(x) = 9x + 8 and g(x) = 3x - 1, find f(x).
A) f(x) = 3x + 11
B) f(x) = 3x - 11
C) f(x) = 3x
D) f(x) = 4x + 11
50) If (f∘g)(x) = 2 2x - 1 and f(x) = x + 8, find g(x).
A) g(x) = 9x - 12
B) g(x) = 8x + 12
C) g(x) = 8x - 12
D) g(x) = 8x
51) If (f∘g)(x) = 16x + 1 and f(x) = 4x + 5, find g(x).
A) g(x) = 4x
B) g(x) = 4x - 1
C) g(x) = 4x + 1
D) g(x) = 3x - 1
52) If (f∘g)(x) = x - 2 and g(x) =
A) f(x) = x2 - 6
C) f(x) = x2
D) f(x) = x2 + 6
53) If (f∘g)(x) =
A) g(x) =
x - 8, find f(x).
B) f(x) = x2 +2x + 6
1
and f(x) = 1 , find g(x).
x-7
x-7
x
B) g(x) =
x+4
C) g(x) =
2x - 4
D) g(x) =
2x
54) The table shows the mean annual compensation of elementary school teachers. Use the linear regression
equation for the data to predict the teachers' average annual compensation in 2018.
Year
1980
1984
1986
1990
1993
1995
1999
2002
A) $50,067
Annual Compenstaion
(dollars)
15,738
18,122
20,049
24,813
27,295
29,367
32,724
34,981
B) $48,229
C) $50,985
10
D) $49,148
55) The table shows the total stopping distance of a sport utility vehicle as a function of its speed. Find the
quadratic regression equation for the data.
Speed (mph)
20
25
30
35
40
45
50
55
60
65
70
75
80
Average total stopping
distance (ft)
47
61.5
78.5
97
122
149
180
216.5
254
298.5
350
408
473
A) y = 0.0890x2 + 1.9615x + 54.9201
C) y = 0.0890x - 1.9615
B) y = 0.0890x2 - 1.9615x + 54.9201
D) y = 54.9201x2 - 1.9615x + 0.0890
56) A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 11 inches
by 30 inches by cutting out equal squares of side x at each corner and then folding up the sides as in the
figure. Express the volume V of the box as a function of x.
30
11
A) V(x) = x(11 - 2x)(30 - 2x)
C) V(x) = (11 - 2x)(30 - 2x)
B) V(x) = (11 - x)(30 - x)
D) V(x) = x(11 - x)(30 - x)
11
57) The figure shown here shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units
long. Express the area A of the rectangle in terms of x.
-1
1
A) A(x) = x(1 - x)
B) A(x) = 2x2
C) A(x) = 2x(x - 1)
D) A(x) = 2x(1 - x)
58) A cone is constructed from a circular piece of paper with a 4-inch radius by cutting out a sector of the circle
with arc length x. The two edges of the remaining portion are joined together to form a cone with radius r
and height h, as shown in the figure. Express the volume V of the cone as a function of x.
4 in.
4 in.
2
A) V(x) = (8π - x)(16πx - x )
24π2
2 16πx - x2
B) V(x) = (8π - x)
24π2
2
C) V(x) = (8π - x)
2
2
D) V(x) = (8π - x) (16πx - x )
4π2
16πx - x2
4π2
12
59) A power plant is located on a river that is 650 feet wide. To lay a new cable from the plant to a location in a
city 2 miles downstream on the opposite side costs $200 per foot across the river and $150 per foot along the
land. Suppose that the cable goes from the plant to a point Q on the opposite side that is x feet from the point
P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the
distance x.
2 mi
650 ft
A) C(x) = 200
x2 + 6502 + 150(10,560 - x)
B) C(x) = 150
C) C(x) = 200
x2 + 6502 + 150(2 - x)
D) C(x) = 200(650 - x) + 150(2 - x)
Rewrite the exponential expression to have the indicated base.
60) 92x; base 3
A) 3-4x
B) 34x
x2 + 6502 + 200(10,560 - x)
C) 35x
D) 38x
C) 3-5x
D) 34x
2x
61) 1
; base 3
9
A) 3-8x
B) 3-4x
Use your grapher to find the zero of the function. Round your answer to three decimal places.
62) f(x) = ex - 2
A) 0.803
B) 0.693
C) 1.993
63) f(x) = 6 - 2x
A) 2.566
B) 2.551
C) 2.585
D) 0.592
D) 2.61
Solve the problem.
64) A certain radioactive isotope has a half-life of approximately 1250 years. How many years to the nearest
year would be required for a given amount of this isotope to decay to 40% of that amount?
A) 1652 years
B) 921 years
C) 750 years
D) 1612 years
65) How long will it take for the population of a certain country to double if its annual growth rate is 5.8%?
Round to the nearest year.
A) 12 years
B) 1 year
C) 5 years
D) 34 years
66) There are currently 80 million cars in a certain country, decreasing by 4.7% annually. How many years will it
take for this country to have 51 million cars? Round to the nearest year.
A) 72 years
B) 3 years
C) 10 years
D) 6 years
13
67) Assume the cost of a car is $32,000. With continuous compounding in effect, find the number of years it
would take to double the cost of the car at an annual inflation rate of 5.1%. Round the answer to the nearest
hundredth.
A) 13.59
B) 203.40
C) 2.03
D) 216.99
68) The population of a small country increases according to the function B = 2,300,000e0.04t, where t is
measured in years. How many people will the country have after 2 years?
A) 2,765,208
B) 2,491,560
C) 5,809,176
D) 2,522,893
Determine if the function is one-to-one.
69)
y
10
5
-10
-5
5
10
x
-5
-10
A) Yes
B) No
70)
y
10
5
-10
-5
5
10
x
-5
-10
A) No
B) Yes
71)
y
10
5
-10
-5
5
10
x
-5
-10
A) Yes
B) No
14
72)
y
10
5
-10
-5
5
10
x
-5
-10
A) No
B) Yes
Find the inverse of the function.
73) f(x) = 4x + 9
2x + 8
A) f-1(x) = 2x - 4
-8x + 9
B) f1(x) = -8x + 9
2x - 4
C) f-1(x) = 4x + 9
2x + 8
D) Not a one-to-one function
74) f(x) =
6
x+5
A) f-1(x) =
75) f(x) =
x
5 + 6x
B) f-1(x) = 5 + 6x
x
C) f-1(x) = -5x + 6
x
D) Not invertible
5x + 9
2
A) f-1(x) = x - 9 for x ≥ 0
5
2
B) f-1(x) = (x - 9)
5
C) f-1(x) = 2x - 9
5
D) f-1(x) = x2 - 9 for x ≥ 0
5
for x ≥ 0
76) f(x) = 3 x - 2
6
A) f-1(x) = 6(x + 2)3
C) f-1(x) = [6(x + 2)]3
77) f(x) = x 2 + 8, x ≥ 0
A) f-1(x) = - x - 8
B) f-1(x) = 18(x + 2)
D) f-1(x) = 6(x3 + 2)
B) f-1(x) =
x-8
15
C) f-1(x) =
x+8
D) f-1(x) =
x-8
78) f(x) = 2x + 6
A) f-1(x) = x + 6
2
B) f-1(x) = x - 6
2
C) Not a one-to-one function
D) f-1(x) = x - 6
2
79) f(x) = x 3 - 1
3
A) f-1(x) = x + 1
B) Not a one-to-one function
3
C) f-1(x) = x + 1
3
D) f-1(x) = x - 1
80) f(x) = 3x3 - 5
3 x-5
B) f-1(x) =
3
A) f-1(x) = 3 x + 5
3
C) f-1(x) =
3 x+5
3
D) Not a one-to-one function
Solve the equation.
81) (5.53)t = 14
A) t = ln 13
ln 5.53
B) t =
14
ln 5.53
C) t = ln 14
ln 5.53
D) t = - ln 14
ln 5.53
C) t = 0.95
ln 3
D) t = 0.95 ln 3
82) e0.95t = 3
A) t = ln 3
0.95
B) t = ln 4
0.95
83) 3x + 3-x = 5
A) x = log3 7 ± 7
4
B) x = log3
C) x = log3 5 ±
D) x = log 5
21
2
21
2
21
2
84) ex + e-x = 9
A) x = ln 9 ±
B) x = - ln 9 ±
77
2
C) x = - ln (9 ±
D) x = ln (9 ±
77)
85) ln y = 7t - 2 ; Solve for y.
A) y = e7t - 2
C) y = e-7t + 2
B) y = 29t
16
77
2
77)
D) y = - e8t - 1
86) ln (y + 4) - ln 8 = x + ln x ; Solve for y.
A) y = 8xex - 4
B) y = 10xe-x - 5
C) y = xe4x + 8
D) y = xe8x + 4
Solve the problem.
87) How long will it take for the population of a certain country to triple if its annual growth rate is 2.5%?
Round to the nearest year.
A) 19 years
B) 44 years
C) 1 year
D) 120 years
88) How long will it take for prices in the economy to double at a 12% annual inflation rate?
A) 23.45 years
B) 5.67 years
C) 6.12 years
D) 9.69 years
89) A college student invests $10,000 in an account paying 14% per year compounded annually. In how many
years will the amount double?
A) 7 years
B) 8.4 years
C) 5.3 years
D) 9.6 years
90) Find a natural logarithmic regression equation for the following data and use it to estimate the production
level for the year 1975. (For the regression equation, assume t = 0 is the year 1950.)
Production
Year (in millions)
1960 9.26
1970 51.43
1990 68.91
A) 50.2
B) 203.449167
C) 52.8
D) 66.8291429
91) Estimate the y-value associated with x = 35 as predicted by the natural logarithmic regression equation for
the following data.
x
10
20
30
40
y
1.51
2.88
3.71
4.36
A) 3.75
B) 3.55
C) 4.05
D) 4.25
Find the requested function value meeting all of the given conditions.
92) tan θ = - 1 and sin θ > 0; Find cos θ.
A) -
2
B)
2
3
C)
2
2
2
D) -
Find the exact value of the real number y.
93) y = sin-1
A) 3π
4
3
2
B) π
3
C) π
4
17
D) 2π
3
3
2
94) y = cos-1
2
2
A) π
6
B) 7π
4
C) 11π
6
D) π
4
95) y = cot-1 (-1)
A) 7π
4
B) 3π
4
C) π
4
D) 5π
4
96) y = sin-1 (-0.5)
A) π
3
B) - π
3
C) - π
6
D) π
6
97) y = arctan 1
A) π
3
B) 2π
3
C) π
4
D) 3π
4
B) -π
C) π
D) π
2
B) π
6
C) π
D) - π
6
B) 3π
4
C) - π
4
D) π
4
B) π
C) 2π
D) π
4
B) π
4
C) π
3
D) - π
4
98) y = arcsec (1)
A) 0
99) y = arcsin - 1
2
A) 4π
3
2
100) y = arccos -
2
A) 7π
4
101) y = csc-1(-1)
A) - π
2
102) y = sin-1
A) π
2
2
2
18
Complete the identity.
103) sec x - 1 = ?
sec x
A) -2 tan2 x
B) 1 + cot x
C) sec x csc x
D) sin x tan x
B) sec x csc x
C) -2 tan2 x
D) 1 + cot x
B) sin x tan x
C) -2 tan2 x
D) 1 + cot x
B) 0
C) 1
D) 1 - sin x
B) 1
C) 1 - sin x
D) 0
B) 1
C) 0
D) 1 - sin x
B) sin2 x + 1
C) cot2 x - 1
D) 1
B) sec2 x
C) tan2 x
D) cot3 x
B) cot2 x
C) sec2 x
D) csc2 x
B) sec4 x + 2
C) 4 sec4 x
D) tan2 x - 1
B) sin x
C) 1
D) 0
C) cot x
D) 1
104) csc x(sin x + cos x) = ?
A) sin x tan x
105) sin x + cos x = ?
cos x sin x
A) sec x csc x
106)
(sin x + cos x)2 = ?
1 + 2 sin x cos x
A) - sec2 x
107) 2 tan x - (1 + tan x)2 = ?
A) - sec2 x
108) tan x(cot x - cos x) = ?
A) - sec2 x
109) sin2 x + sin2 x cot2 x = ?
A) cot2 x + 1
110) sin2 x + tan2 x + cot2 x = ?
A) sin x
111) csc x cot x = ?
sec x
A) 1
112) sec4 x + sec2 x tan2 x - 2 tan4 x = ?
A) 3 sec4 x - 2
113) cot x · tan x = ?
A) -1
114)
(cot x + 1)(cot x + 1) - csc2 x = ?
cot x
A) 0
B) 2
19
115) sin x - cos x + cos x - sin x = ?
sin x
cos x
A) sec x csc x
B) 2 + sec x csc x
C) 2 - sec x csc x
D) 1 - sec x csc x
B) -1
C) sin2 x
D) cos2 x
B) 2 - sec x csc x
C) 2 + sec x csc x
D) sec x csc x
B) 1 + 2sin2 x
C) 1 - 2cos2 x
D) 1 - 2sin2 x
B) 1
C) 2
D) 0
C) sec2 2x
D) 2
B) 0
C) cot x
D) sin x
B) csc x + cot x
C) csc x - cot x + 1
D) csc x - cot x
B) sec x + csc x
C) sec 2 x + csc 2 x
D) csc 2 x - sec 2 x
B) cot α + tan β
C) tan β + 1
D) cot α + tan α
2
2
116) sin x - cos x = ?
1 - cot2 x
A) 1
117) cos x + sin x - sin x - cos x = ?
cos x
sin x
A) 1 - sec x csc x
118) cos4 x - sin4 x = ?
A) 1 + 2cos2 x
119)
(sec x + 1)(sec x - 1) = ?
tan2 x
A) -1
120) tan2 2x + cos2 2x + sin2 2x = ?
A) sin2 2x
B) cos2 2x
121) 1 - cos2 x = ?
1 + sin x
A) tan x
122)
1 - cos x = ?
sin x
A) -csc x - cot x
123) sec2 x csc 2 x = ?
A) sec 2 x - csc 2 x
124)
cos (α - β) = ?
sin α cos β
A) cot α + cot β
20
125) cos x - 5π = ?
6
A) -
2
3 (cos x - sin x)
C) 1 (2
B) -
3 cos x + sin x)
D)
2
2
3 (cos x + sin x)
3 (cos x - sin x)
126) sin x + 3π = ?
2
B) sin x
C) -sin x
D) -cos x
127) sin (α + β) + sin (α - β) = ?
A) 2sin α cos β
B) 2cos α cos β
C) cos β cos α
D) sin α cos β
128) cos (α + β) cos (α - β) = ?
A) cos2 α - sin2 β
B) sin2 β - sin2 α
C) cos2 β - sin2 α
D) cos2 β + sin2 α
B) cot α - cot β
C) tan α - tan β
D) tan β - tan α
A) cos x
129)
sin (α - β) = ?
cos α cos β
A) -tan α - cot β
130)
cos (α - β) = ?
cos (α + β)
A)
1 - tan α tan β
1 + tan (α + β)
B)
1 - tan α tan β
1 + tan α tan β
C)
1 + tan (α + β)
1 - tan α tan β
B)
cos2 2x - cos2 12x
4
D)
1 + tan α tan β
1 - tan α tan β
131) sin 5x sin 7x cos 5x cos 7x = ?
A)
cos2 12x + cos2 2x
4
2
C) sin 70x
4
D) cos2 70x
132) sin x (sin 4x + sin 6x) = ?
A) 1 cos x (cos 4x - cos 6x)
2
B) cos x (cos 4x + cos 6x)
C) 1 cos x (cos 4x + cos 6x)
2
133)
D) cos x (cos 4x - cos 6x)
sin 3x + sin 9x = ?
cos 3x + cos 9x
A) tan 3x + tan 9x
B) tan 6x
C) 2 tan 6x tan 3x
21
D) tan 6x cot 3x
134)
cos 4x - cos 8x = ?
cos 4x + cos 8x
A) -tan 6x
135)
C) cot 6x
D) tan 2x tan 6x
B) sin 12x
cos 10x
C) tan 5 x + tan 7 x
4
6
D) cos 12x
sin 10x
B) cot x - y
2
C) tan x + tan y
D) tan x + y
2
sin 5x + sin 7x = ?
cos 4x + cos 6x
A) sin 6x
cos 5x
136)
B) 0
sin x + sin y = ?
cos x + cos y
A) tan x - y
2
BONUS. EACH TRIG IDENTITY PROOF IS WORTH 20 BONUS POINTS. THEY MUST BE COMPLETED NEATLY ON
A SEPERATE SHEET OF PAPER WITH ALL WORK SHOWN. PROOFS MUST BE CLEAR AND USE PROPER TRIG
IDENTITIES
Verify the identity.
137) cot θ · sec θ = csc θ
138) tan θ · csc θ = sec θ
139) csc2u - cos u sec u= cot2 u
140) (1 + tan2u)(1 - sin2u) = 1
141) csc u - sin u = cos u cot u
142) 1 + sec2x sin2x = sec2x
143) cot 2 x + csc 2 x = 2 csc 2 x - 1
144) cos x + π = -sin x
2
145) sin 3π - θ = -cos θ
2
146) cos 3π - θ = -sin θ
2
22
147) cos(α + β) = cot β - tan α
cos α sin β
x-y
148) cos x + cos y = cot
sin x - sin y
2
149)
sin α - sin β = tan α - β cot α + β
sin α + sin β
2
2
150) sin(α + β) - sin(α - β) = 2 cos α sin β
151) cos(α - β) - cos(α + β) = 2 sin α sin β
152) sin(α - β) cos(α + β) = sin α cos α - sin β cos β
23