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Teacher: Barton
Name: ____________________________________
1. The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a1, which is an
equation for the nth term of this sequence?
1. an= 8n + 10
2. an= 8n - 14
3. an= 16n + 10
4. an= 16n - 38
2. Two forces of 40 pounds and 28 pounds act on an object. The angle between the two forces is 65°. Find the
magnitude of the resultant force, to the nearest pound.
Using this answer, find the measure of the angle formed between the resultant and the smaller force, to the
nearest degree.
3. Solve sec x - = 0 algebraically for all values of x in 0° ≤ x < 360°.
4. A jogger ran mile on day 1, and mile on day 2, and 1 miles on day 3, and 2 miles on day 4, and this
pattern continued for 3 more days. Which expression represents the total distance the jogger ran?
1. 2. 3. 4. 5. What is the period of the graph y = sin 6x?
1. 2. 3. 4. 6π
6. If sin θ < 0 and cot θ > 0, in which quadrant does the terminal side of angle θ lie?
1. I 3. III
2. II 4. IV
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7. What is the common difference of the arithmetic sequence below?
-7x, -4x, -x, 2x, 5x, …
1. -3 3. 3
2. -3x 4. 3x
8. The expression is equivalent to
1. sin x 3. tan x
2. cos x 4. sec x
9. Solve algebraically for all exact values of x in the interval 0 ≤ x < 2π:
2 sin2x + 5 sin x = 3
10. In an arithmetic sequence, a4 = 19 and a7 = 31. Determine a formula for an, the nth term of this sequence.
11. Show that sec θ sin θ cot θ = 1 is an identity.
12. Approximately how many degrees does five radians equal?
1. 286
2. 900
3. 4. 5π
13. Which expression is equivalent to 1. 2a2 + 17
2. 4a2 + 30
3. 2a2 – 10a + 17
4. 4a2–20a + 30
?
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14. If sin A = − and ∠A terminates in Quadrant IV, tan A equals​
1. −
2. −
3. −
4. −
15. What is the common difference in the sequence 2a + 1, 4a + 4, 6a + 7, 8a + 10, …?
1. 2a + 3
2. –2a– 3
3. 2a + 5
4. –2a + 5
16. Express as a single trigonometric function, in simplest form, for all values of x for which it is defined.
17. What is the common ratio of the sequence ?
1. 2. 3. 4. 18. An angle, P, drawn in standard position, terminates in Quadrant II if
1. cos P < 0 and csc P < 0
2. sin P > 0 and cos P > 0
3. csc P > 0 and cot P < 0
4. tan P < 0 and sec P > 0
19. The expression 1. 58 – 4x 3. 58 – 12x
2. 46 – 4x 4. 46 – 12x
is equal to
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20. Which equation represents the graph below?
1. y = –2 sin 2x
2. y = –2 sin x
3. y = –2 cos 2x
4. y = –2 cos x
21. The sum of the first eight terms of the series 3 – 12 + 48 – 192 + … is
1. –13,107 3. –39,321
2. –21,845 4. –65,535
22. A man standing on level ground is 1000 feet away from the base of a 350-foot-tall building. Find, to the nearest
degree, the measure of the angle of elevation to the top of the building from the point on the ground where the
man is standing.
23. Determine the sum of the first twenty terms of the sequence whose first five terms are 5, 14, 23, 32, and 41.
24. What is the solution set of the equation – sec x = 2 when 0° ≤ x < 360°?
1. {45°, 135°, 225°, 315°}
2. {45°, 315°}
3. {135°, 225°}
4. {225°, 315°}
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25. In the diagram below, the length of which line segment is equal to the exact value of sin θ?
1. 2. 3. 4. 26. What is the common ratio of the geometric sequence shown below?
–2, 4, –8, 16, …
1. –
2. 2
3. –2
4. –6
27. a Solve for all values of tan θ to the nearest hundredth:
b Using the answers form part a, find, to the nearest degree, all values of θ which satsify 0° ≤ θ < 360°.
28. The expression is equivalent to
1. cos2 x
2. tan2 x
3. 4. 1 − sin2 x
29. In the equation 2 sin2 x − sin x = 0, angle x may equal
1. 0° 3. 45°
2. 30° 4. 150°
in the interval
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30. The expression sin (−60)° is equivalent to
1. cos 60° 3. sin 60°
2. −cos 60° 4. −sin 60°
31. (a) Sketch and label the graph of y = sin x for values of x in the interval 0 ≤ x ≤ 2π. [3]
(b) On the same set of axes used in part a, sketch and label the graph of y = –cos x for values of x in the
interval 0 ≤ x ≤ 2π. [5]
(c) From the graphs made in parts a and b, find the solution for the equation sin x = –cos x in the
interval 0 ≤ x ≤ 2π. [2]
32. The expression cos 210° is equivalent to:
1. –cos 30° 3. –sin 30°
2. cos 30° 4. sin 30°
33. If sin x = and angle x is in the fourth quadrant, find csc x.
34. Express cos θ (sec θ – cos θ) in terms of sin θ.
35. Which summation represents 5 + 7 + 9 + 11 … + 43?
1. 2. 3. 4. 36. What is the smallest positive value of θ which satisfies the equation 2 cos2 θ – cos θ = 0?
1. 30° 3. 90°
2. 60° 4. 270°
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37. Which function is represented in the accompanying graph over the interval –π ≤ x ≤ π?
1. y = sin x
2. y = cos x
3. y = sin x
4. y = cos x
38. If sec A = and A is an angle in Quadrant IV, find the value of cos A.
1. −
2. 3. −
4. 1
39. Express the sin (–140°) as a function of a positive acute angle.
1. sin 40° 3. – sin 40°
2. tan 40° 4. – tan 40°
40. If sin θ = − and θ lies in Quadrant IV, what is the value of cos θ?
1. 2. −
3. 4. 6/10/2015 10:30:51 AM
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Answer Key for More review
1. 2
2.
Below is a diagram that
represents the given
information in the problem.
The quadrilateral is a
parallelogram. In a parallelogram, the
opposite sides are equal and
the adjacent or consecutive
angles are supplementary.
Below is the diagram with
more information filled in.
The resultant force is
represented as the red
diagonal of the
parallelogram.To find the
magnitude of the force is to
find the distance of the red
line. Use the law of cosines,
c2 = a2 + b2 – 2abcosC, to
find the distance. The value of
a and b are the lengths of the
sides of the parallelogram and
C is the measure of the angle
opposite the red line. C =
115°.
Substitute into the law of
cosines and evaluate.
15. 1
16.
To simplify the expression, rewrite all
the trigonometric functions in terms of
sine and cosine. Rewrite the
expression like this:
29. 1
30. 4
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To the nearest pound the
magnitude is 58. The next part of the question
asks for the measure of the
angle between the resultant
and the smaller force to the
nearest degree. This is
represented by the blue x in
the diagram below. Use the law of sines, to find the measure of x. Let
a = 58, A = 115°, b = 40 and
B = x. Substitute into the law
of sines and solve for x.
To the nearest degree the
answer is 39°.
3.
First solve for sec x:
17. 2
31.
PART A:
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Start with the graph of y = sin x in the
interval 0 ≤ x ≤ 2π, shown in red:
The value of the angle was
positive. Cosine is positive in
quadrants I and IV. The
reference angle in each of the
quadrants is 45°. In quadrant I
the angle is equal to the
reference angle, 45°.
PART B:
Add to that in green the graph of the
equation y = –cos x. The negative in
front of the cosine reflects the graph
over the x-axis and the coefficient of x,
1/ , is the frequency of the equation. 2 In quadrant IV the angle is
equal to the reference angle
subtracted from 360°.
Since the frequency is 1/2 , only half of
the normal cosine curve is seen in the
interval of 0 ≤ x ≤ 2π.
360° - 45° = 315°
x = 45° and 315°
PART C:
The intersection(solution) of the graphs
is the point (π, 0). 4. 1
18. 3
32. 1
5. 2
19. 4
33.
Cosecant, csc, is the reciprocal of sine. If sine is , then cosecant is .
6. 3
20. 3
34.
When performing operations with
trigonometric functions, first express
all the functions in terms of either
cosine or sine. Secant is the reciprocal
of cosine.
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Substitute the fraction for sec θ. In order to simplify this into terms of
sin θ, use the Pythagorean identity sin2
θ + cos2 θ = 1. If we subtract cos2 θ
from both sides of the equation, the
result is sin2 θ = 1 − cos2 θ. So in
terms of sin θ, we have:
7. 4
21. 3
35. 2
8. 2
22.
36. 2
Below is a diagram of the information
provided in the question. Use right
triangle trigonometry to find the value
of x, SohCahToa. Below is the diagram with the sides
labeled for the trigonometry.
Use the tangent ratio, opposite over
adjacent, to find the value of x.
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To the nearest degree, x = 19°.
9.
23.
Set the equation equal to
zero. Then use the quadratic
formula to solve for sin x. The very first thing that must be
established is whether the series is
arithmetic or geometric.
2 sin2x + 5 sin x = 3
2 sin2x + 5 sin x – 3 = 0
The series is arithmetic because there
is an increase of 9 from one term to
the next. The common difference is 9.
Apply the quadratic formula
with a = 2, b = 5 and c = –3. Substitute the values into the
formula and simplify.
Now use arcsine to find the
angle measure For the values
of sin x.
To find the sum of the first twenty
terms, we need to know both the first
term, a1, and the twentieth term, a20,
of the series. Use the general term
formula for an arithmetic sequence,
which is an = a1 + (n − 1)d, where a1
is the first term of the sequence, n is
the number of the term we are trying
to find, and d is the common
difference. The first term of the
sequence is a1 = 5. We are looking for
the twentieth term, so n = 20 and d =
9. Substitute those values into the
formula and evaluate.
Now find the sum of the arithmetic
series using the formula: .
Remember that n = 20 and that a20 =
The value of sine cannot be
greater than 1 or less than –1. 176. Substitute into the formula and
simplify.
In particular, sine cannot be
equal to –3. Reject that
possibility. Sine is equal to 30° in
quadrants I and II, because
1/ is positive and sine is
2
positive in quadrants I and II.
In quadrant I, a reference
angle of 30° results in an
angle of 30°.
In quadrant II, a reference
37. 1
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angle of 30° results in an
angle of 180° – 30° =
150°.
Lastly, since the interval was
given in radians the answers
must also be expressed in
radians. Convert 30° and
150° into radian measure by
multiplying by π/180.
10.
The general term of an
arithmetic sequence is in the
form of an = a1 + (n – 1)d,
where a1 represents the first
term in the sequence, n
represents the term number,
and d is the common
difference. To determine a
formula for an, the nth term of
this sequence, first fine the
values for a1 and d. To find d, subtract the
difference between the term
values, 31 – 19 = 12, and
divide that by the difference
in the term numbers, 7 – 4 =
3.
d = 12/3 = 4
To find the first term, a1,
subtract 4 three times from
the fourth term, a4. 19 – 4 = 15
15 – 4 = 11
11 – 4 = 7
The first term of the sequence
is a1 = 7.
24. 3
38. 2
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The formula for the sequence
can be written as​ an = 7 + (n
– 1)4 or an equivalent
equation.
11.
25. 2
39. 3
12. 1
26. 3
40. 1
13. 4
27.
Rewrite each of the
trigonometric ratios into
terms of sine and cosine.
Multiply the three ratios
together.
PART (A):
First, multiply through by tan θ to get
rid of the fractions and to form an
equivalent quadratic equation. Be sure
to put it in standard form.
Next, solve for θ using the quadratic
formula with a = 1, b = –6 and c = 1. Substitute the values into the formula
and simplify. To the nearest hundredth, tan θ = 5.83
or 0.17
PART (B):
To find the value of θ, use the arctan
function, tan–1. tan–1(5.83) = 80.26696601°
tan–1(0.17) = 9.648°
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Both values of tangent are positive. Tan is positive in quadrants I and III. In quadrant I, the value of the angle is
simply equal to the reference angle. To
the nearest degree, θ = 80° or 10° in
quadrant I. In quadrant III, add 180° to each
reference angle to find the value of θ:
180° + 80° = 260° and 180° + 10° =
190°
14. 2
28. 3