Solving Trigonometric Equations — Cumulative exam Answers
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1
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What is the value of 1 − 2sin2(105°)?
A
Negative StartFraction StartRoot 3 EndRoot Over 2 EndFraction
B
Negative one-half
C
One-half
D
StartFraction StartRoot 3 EndRoot Over 2 EndFraction
2
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Let sin(2x) = cos(x), where 0° ≤ x < 180°. What are the possible values for x?
A
30° onlyB
90° onlyC
30° or 150°D
30°, 90°, or 150°3
Judy knows that the function graphed below is a transformation of the parent function . Since the period and asymptotes are the same as those of the parent function, she knows that there were no horizontal stretches, compressions, or translations. She can also tell that there were no reflections since the general shape of the graph is the same as that of the parent function. Which statement is correct regarding the transformations that were applied?
A
Since the y-intercept of the graph is , the graph of the parent function was translated 0.5 units down, and since the graph passes through the point instead of , the graph was vertically compressed.
B
Since the y-intercept of the graph is , the graph of the parent function was translated 0.5 units down, and since the graph passes through the point instead of , the graph was vertically stretched.
C
Since the y-intercept of the graph is , the graph of the parent function was translated 1.5 units down, and since the graph passes through the point instead of , the graph was vertically compressed.
D
Since the y-intercept of the graph is , the graph of the parent function was translated 1.5 units down, and since the graph passes through the point instead of , the graph was vertically stretched.
4
Which point does the graph of the parent function pass through?
A
(StartFraction square root of 3 over 3 EndFraction ,StartFraction pi over 3 EndFraction)
B
(StartFraction pi over 3 EndFraction , StartFraction square root of 3 over 3 EndFraction )
C
(StartFraction pi over 3 EndFraction , square root of 3)
D
( square root of 3 , StartFraction pi over 3 EndFraction)
5
Which is the graph of ?
A
A smooth curve resembling an arctangent function. The x-axis ranges from -6 to 6, and the y-axis from -p to p. The curve approaches horizontal asymptotes at y = p/2 and y = -p/2, passing through the origin.
B
A graph of a function with asymptotic behavior. The x-axis ranges from -5 to 5, and the y-axis ranges from -p to p. The curve is located in the fourth quadrant, approaching the y-axis asymptotically as x approaches zero from both positive and negative directions. The curve also approaches zero asymptotically as x moves away from zero in both directions. The function does not intersect either axis.
C
A graph of a function with the x-axis ranging from -2.5 to 2.5 and the y-axis ranging from -p to p. The curve passes through the origin (0,0) and changes direction at y=0. It intersects the x-axis at two points, approximately at x=-1 and x=1. The curve is symmetric about the origin.
D
The image shows a mathematical graph plotted on a Cartesian coordinate system. The x-axis ranges from -2 to 2, and the y-axis ranges from -p to p. A smooth, continuous curve intersects the x-axis at approximately 1. The curve extends into both the positive and negative quadrants. The point p is labeled on the positive y-axis and -p on the negative y-axis.
6
Read the proof.
A
tangent double angle identity
B
tangent sum identityC
tangent double angle identity
D
tangent sum identity7
Which statement is true about the graph of the equation ?
A
There is a horizontal asymptote at .
B
There is a horizontal asymptote at .
C
There is a vertical asymptote at .
D
There is a vertical asymptote at .
8
Solve the Law of Cosine: for cos C.
A
cosine C = StartFraction square of a + square of b minus square of c over 2ab EndFraction
B
cosine C = StartFraction square of a + square of b + square of c over negative 2ab EndFraction
C
cosine C = StartFraction square of c minus square of a minus square of b over 2ab EndFraction
D
cosine C = StartFraction square of negative c + square of a + square of b over negative 2ab EndFraction
9
What is the exact value of tan ?
A
Negative 2 minus StartRoot 3 EndRoot
B
Negative 2 + StartRoot 3 EndRoot
C
1 minus StartRoot 3 EndRoot
D
1 + StartRoot 3 EndRoot
10
Which is the graph of the function ?
A
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 4 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 2. The curve crosses the x axis at (pi comma 0), (0 comma 0) and (negative pi comma 0).
B
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 4 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 2. The curve crosses the x axis at (pi comma 0), (0 comma 0) and (negative pi comma 0).
C
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 4 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 4. The curve crosses the x axis at (pi by 2 comma 0) and (negative 3 pi by 4 comma 0). The y-intercept is at (0 comma negative 2).
D
On a coordinate plane, the x axis ranges from negative pi to pi with an interval of pi by 4 units and the y axis ranges from negative 6 to 6 with an interval of 2 units. It shows a curve that oscillates between positive and negative infinity as x approaches multiples of pi. The curve has vertical asymptotes at x equals plus or minus n times pi by 4. The curve crosses the x axis at (pi by 2 comma 0) and (negative 3 pi by 4 comma 0). The y-intercept is at (0 comma negative 1).
11
What is the domain of the function ?
A
all real numbers except , where n is any integer
B
all real numbers except , where n is any integer
C
all real numbers except , where n is any integer
D
all real numbers except , where n is any integer
12
What is the exact value of sin(105°)?
A
Negative StartFraction StartStartRoot 2 minus StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
B
Negative StartFraction StartStartRoot 2 + StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
C
StartFraction StartStartRoot 2 minus StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
D
StartFraction StartStartRoot 2 + StartRoot 3 EndRoot EndEndRoot Over 2 EndFraction
14
For 0 ≤ x < 2π, which of the following solves 2cos + 1 = 0?
A
Left-brace StartFraction 4 pi Over 3 EndFraction right-brace
B
Left-brace StartFraction 5 pi Over 3 EndFraction right-brace
C
Left-brace StartFraction pi Over 3 EndFraction, StartFraction 2 pi Over 3 EndFraction right-brace
D
Left-brace StartFraction pi Over 3 EndFraction, StartFraction 5 pi Over 3 EndFraction right-brace
15
Which expression is equivalent to ?
A
Cosine (negative StartFraction pi Over 3 EndFraction)
B
Sine (negative StartFraction pi Over 3 EndFraction)
C
Cosine (StartFraction pi Over 2 EndFraction)
D
Sine (StartFraction pi Over 2 EndFraction)
16
Which equation could be used to solve for the measure of side q in shown below, given s = 15, S = 45°, and Q = 105°? [Not drawn to scale]
A
StartFraction sine 105 degree over 15 EndFraction = StartFraction sine 30 degree over q EndFraction
B
StartFraction sine 105 degree over q EndFraction = StartFraction sine 30 degree over 15 EndFraction
C
StartFraction 15 over sine 105 degree EndFraction = StartFraction q over sine 45 degree EndFraction
D
StartFraction q over sine 105 degree EndFraction = StartFraction 15 over sine 45 degree EndFraction
17
When deriving the Law of Cosines, the equation is created using the Pythagorean theorem. Use substitution to replace in the equation below and then simplify.
A
square of c = square of a + square of b minus 2bx
B
square of c = square of a minus square of b minus 2bx
C
square of c = square of a minus square of x + square of b minus 2bx + square of x
D
square of c = square of h + square of a + square of b minus 2bx
18
Which shows a correct simplification of sin(x + )?
A
sin(x + )= sin(x)cos() + cos(x)sin()= sin(x) · 1 + cos(x) · 0= sin(x)B
sin(x + )= cos(x)cos() – sin(x)sin()= cos(x) · 1 – sin(x) · 0= cos(x)C
sin(x + )= sin(x)cos() + cos(x)sin()= sin(x) · –1 + cos(x) · 0= –sin(x)D
sin(x + )= cos(x)cos() – sin(x)sin()= cos(x) · –1 – sin(x) · 0= –cos(x)19
What restriction should be applied to y = tanx for y = arctanx to be defined?
A
Restrict the range to (negative StartFraction pi Over 2 EndFraction, StartFraction pi Over 2 EndFraction)
B
Restrict the range to left-bracket negative StartFraction pi Over 2 EndFraction, StartFraction pi Over 2 EndFraction right-bracket
C
Restrict the domain to (negative StartFraction pi Over 2 EndFraction, StartFraction pi Over 2 EndFraction)
D
Restrict the domain to left-bracket negative StartFraction pi Over 2 EndFraction, StartFraction pi Over 2 EndFraction right-bracket
20
What is the exact value of tan(195°)?
A
StartFraction StartRoot 3 EndRoot + 1 Over 1 minus StartRoot 3 EndRoot EndFraction
B
StartFraction StartRoot 3 EndRoot minus 3 Over 3 + StartRoot 3 EndRoot EndFraction
C
StartFraction StartRoot 3 EndRoot minus 1 Over 1 + StartRoot 3 EndRoot EndFraction
D
StartFraction StartRoot 3 EndRoot + 3 Over 3 minus StartRoot 3 EndRoot EndFraction
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