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Trig/calc > practice problems > Flashcards

practice problems Flashcards

1
Q

Convert to degree measure: πœ‹/12

A

15Β°

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2
Q

Convert to decimal degree. Round to the nearest hundredth, if necessary: 256Β° 3β€² 23”

A

256.06Β°

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3
Q

Convert to DMS: 60.98Β°

A

60Β° 58β€²48”

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4
Q

Find the reference angle and then give the exact value of sin 7πœ‹/6

A

The reference angle is πœ‹/6,
sin 7πœ‹/6 = ― 1/2

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5
Q

Find tan πœƒ, given sec πœƒ = 5
2 and that πœƒ is in quadrant IV.

A

-sqrt21/2

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6
Q

A weight is attached to a rope on a pully with radius 9.29 inches. How many inches will the weight rise if the pully is rotated through
an angle of 78Β° 40β€². Round to the nearest tenth.

A

12.8 inches

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7
Q

Find the angle of least possible positive measure coterminal with ― 11πœ‹\3

A

πœ‹/3

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8
Q

The equation of the terminal side of an angle πœƒ is 3π‘₯ + 𝑦 = 0, π‘₯ ≀ 0. Find the value of sec πœƒ

A

sec πœƒ = ―sqrt 10

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9
Q

From a boat on a lake, the angle of elevation to the top of a cliff is 13Β° 53β€². If the base of the cliff is 1217 feet from the boat, how
high is the cliff (to the nearest foot)?

A

301 ft

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10
Q

Identify the quadrant that satisfies the following conditions: cot πœƒ < 0 and cos πœƒ > 0. Quadrant #

A

IV

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11
Q

Solve the triangle below. Round to two decimal places.
a=29.43cm, c=53.58cm, C=90Β°

A

𝑏 β‰ˆ 44.77 cm, 𝐴 β‰ˆ 33.32Β°, 𝐡 β‰ˆ 56.68Β°

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12
Q

Give a piecewise function that describes the graph below: (graph shown in #12a)

A

𝑓(π‘₯) = { π‘₯ π‘₯ ≀ 1
{―π‘₯ + 3 π‘₯ > 1

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13
Q

Sketch a comprehensive graph of 𝑃(π‘₯) = (2π‘₯ ― 3)(π‘₯ + 3)2.

A

(answer shown in #13a)

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14
Q

Solve the polynomial inequality. Give your answer in interval notation!
(π‘₯ + 3)^2(π‘₯ ― 7) β‰₯ 0

A

{ ―3} βˆͺ [7, ∞)

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15
Q

Graph the function 𝑓(π‘₯) = 2π‘₯4 + π‘₯3 ―14π‘₯2 ―9π‘₯ ― 36. A good viewing window is [-6, 6] by [-90, 10].
Give the domain and range

A

Domain: ( β€•βˆž,∞ )
Range: [ ― 70.81, ∞)

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16
Q

Graph the function 𝑓(π‘₯) = 2π‘₯4 + π‘₯3 ―14π‘₯2 ―9π‘₯ ― 36. A good viewing window is [-6, 6] by [-90, 10].
List the intervals over which the function is increasing and decreasing

A

Increasing: ( ―1.90, ― 0.32) βˆͺ (1.85, ∞)
Decreasing: (∞, ― 1.90) βˆͺ ( ― 0.32, 1.85)

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17
Q

Given that π‘˜ = ―7 is a zero of 𝑃(π‘₯) = 3π‘₯3 +16π‘₯2 ―33π‘₯ + 14, factor 𝑃(π‘₯) completely into linear factors.

A

𝑃(π‘₯) = (π‘₯ + 7)(π‘₯ ― 1)(3π‘₯ ― 2)

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18
Q

Find a function, 𝑃(π‘₯), defined by a polynomial of degree 4 with real coefficients that has zeros 𝑖 and 3𝑖 and where 𝑃( ―1) = 40.

A

𝑃(π‘₯) = 2π‘₯4 + 20π‘₯2 + 18

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19
Q

Given that 2𝑖 is a zero of 𝑓(π‘₯) = π‘₯4 ― π‘₯3 +2π‘₯2 ―4π‘₯ ― 8, find all the other zeros.

A

The zeros are: Β± 2𝑖, ― 1, 2

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20
Q

Use the Rational Zeros Theorem to find all the zeros of 𝑃(π‘₯) = 3π‘₯3 ―8π‘₯2 ―8π‘₯ + 8

A

The zeros are: 2/3, 1 Β± sqrt5

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21
Q

Sketch a graph of 𝑦 = ―2cos (π‘₯ + πœ‹/2). Include the table with your β€œfive important points” from the same period

A

(answer shown in #1)

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22
Q

Sketch a graph of 𝑦 = ―1 + 3cot (1
2π‘₯). Include the table with your β€œthree important points” and two asymptotes from the same
period.

A

(answer shown in #2)

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23
Q

List the amplitude, period, range, and phase shift of: 𝑦 = 1 + 4sin (3π‘₯ ― πœ‹).

A

Amplitude: 4, Period: 2πœ‹/3 , Range: [ ― 3, 5], Phase Shift: Right πœ‹/3

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24
Q

List the period, range, and phase shift of: 𝑦 = ―2tan [4(π‘₯ + πœ‹/4)]

A

Period: πœ‹/4, Range: ( ― ∞, ∞), Phase Shift: Left πœ‹/4

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25
Q

The position of a weight attached to a spring is 𝑠(𝑑) = ―4cos (20πœ‹π‘‘) inches after 𝑑 seconds. What is the maximum height that the
weight reaches above the equilibrium position and when does it reach the fist maximum? Include units

A

Max Height: 4 inches, Reaches First Max: at 1/20 second

26
Q

A weight attached to a spring is pulled down 2 inches below the equilibrium position. Assuming that the period of the system is 1
6
second, determine a trigonometric model that gives the position of the weight at time 𝑑 seconds.

A

𝑠(𝑑) = ―2cos (12πœ‹ 𝑑)

27
Q

Verify the identity: (show in #7a)

A

(shown in 7a)

28
Q

Use a sum/difference identity to find the exact value of the expression. Hint: Use an even/odd identity to simplify.
cos ( ― 7πœ‹/12)

A

sqrt2-sqrt6
_____________
4

29
Q

Find the exact value by using a sum/difference identity: sin 95Β°cos 35Β° ― cos 95Β°sin 35Β°

A

sqrt3/2

30
Q

Given that cos 𝐢 = ― 5
13 and 𝐢 is in quadrant II, find sin 2𝐢

A

-120/169

31
Q

Given that cos 𝐢 = ― 5
13 and 𝐢 is in quadrant II, find cos 2𝐢.

A

-119/169

32
Q

Find the exact value by using a half-number identity: sin 15πœ‹/8

A

shown in 12b

33
Q

Graph the function by using transformations: ― 2π‘₯ +3. Be sure to label your axes or list at least three points on the final graph

A

shown in 13c

34
Q

use 𝐴 = 𝑃(1 + π‘Ÿ/𝑛)^𝑛^𝑑 or 𝐴 = 𝑃𝑒^π‘Ÿ^𝑑
Determine the amount of money in an account after 5 years if $1000 is deposited and earns 8% compounded monthly. Round to
the nearest cent.

A

$1489.85

35
Q

use 𝐴 = 𝑃(1 + π‘Ÿ/𝑛)^𝑛^𝑑 or 𝐴 = 𝑃𝑒^π‘Ÿ^𝑑
If $2000 is invested in an account that pays interest compounded continuously, how long will it take to grow to $3500 if the interest
rate is 6%? Round to the nearest tenth

A

9.3 years

36
Q

Write an equivalent expression in logarithmic form: 3^ ―2 = 1/9

A

shown in 15d

37
Q

Use the change-of-base formula to estimate to four decimal places: log8 14.23

A

1.2770

38
Q

Find the domain of the function. Give your answer in interval notation (shown in 18e)

A

(-7,7)

39
Q

Solve the equation. Give your solution(s) in exact form. 𝑒^2^π‘₯ ― 𝑒^π‘₯ ―6 = 0.

A

x=ln3

40
Q

Solve the equation. Be sure to check for extraneous solutions! (shown in 20f

A

x=1

41
Q

State the range of the inverse cotangent function

A

(0, πœ‹)

42
Q

State the range of the inverse sine function

A

[ ― πœ‹/2,πœ‹/2]

43
Q

Find the exact value (in radian) of csc ―1 ( ― 2)

A

-πœ‹/6

44
Q

Find the value (in degrees) of
cot ^-1 (-sqrt 3)

A

150Β°

45
Q

Find the exact value of sin (arctan 2)

A

2 sqrt5/5

46
Q

Solve for all solutions in the interval [0, 2πœ‹): 4tan^2 π‘₯ ―12 = 0

A

{ πœ‹/3,2πœ‹/3 ,4πœ‹/3 ,5πœ‹/3 }

47
Q

Solve for all solutions in the interval [0, 2πœ‹): 2sin2 π‘₯ ― sin π‘₯ > 0
Give your answer in interval notation with exact values for the endpoints.

A

(πœ‹/6,5πœ‹/6 ) βˆͺ (πœ‹, 2πœ‹)

48
Q

Solve for all solutions in the interval [0Β°, 360Β°): sin 2πœƒ = ― 1/2

A

{105Β°, 165Β°, 285Β°, 345Β°}

49
Q

Solve for all solutions in the interval [0, 2πœ‹): cos 2π‘₯ < ― 2
2
Give your answer in interval notation with exact values for the endpoints

A

(3πœ‹/8 ,5πœ‹/8 ) βˆͺ (11πœ‹/8 ,13πœ‹/8 )

50
Q

Give the equation of the oblique asymptote of 𝑓(π‘₯)
x^2-7x+3
————–
x+3

A

𝑦 = π‘₯ ― 10

51
Q

Sketch a comprehensive graph of 𝑓. Include the correct end behavior, all asymptotes, and intercepts and approximations of the
locations of the turning points.
f(x)= x-1
———–
x^2+x-20

A

(shown in 11a)

52
Q

Solve the rational inequality. Express your solution in interval notation
(shown in 12b)

A

( β€•βˆž, 3] βˆͺ (4, 6)

53
Q

The volume of a gas varies inversely as its pressure and directly as its temperature. If a certain gas occupies a volume of 2.2 L at a
temperature of 360 K and a pressure of 20 N/cm2, find the volume when the temperature is 324 K and the pressure is 30 N/cm2..

A

1.32 L

54
Q

Solve the system. Express your solution in terms of 𝑧.
{π‘₯ + 3𝑦 + 2𝑧 = 11
{4𝑦 + 9𝑧 = ―12

A

(19𝑧/4 + 20, ― 9/4𝑧 ― 3, 𝑧)

55
Q

Solve the system. Use an augmented matrix. You must show your work for credit.
{ 2π‘₯ ― 2𝑦 ― 𝑧 = ―4
{π‘₯ ― 4𝑦 ― 5𝑧 = ―44
{―9π‘₯ + 𝑦 + 𝑧 = ―60

A

(8,8,4)

56
Q

Suppose you are solving a system of three linear equations by the row echelon method and obtain the following matrix:
[1 12 ―8 |21]
[0 0 0 |0 ]
[0 10 ―3 |25]
What conclusion can you draw about the solution(s) of this system? (Choose one).
(a) The system has no solutions.
(b) The system has exactly one solution.
(c) The system has infinitely many solutions.
(d) There is not enough information to draw any conclusions about the solution(s) of the system.

A

c

57
Q

Find the determinant:
[4 3 5]
det [5 3 5]
[3 3 3]

A

6

58
Q

Solve the system using Cramer’s Rule. You must show your work for credit.
{ π‘₯ + 8𝑦 = ―47
{―3π‘₯ + 7𝑦 = ―45

A

(1, ― 6)

59
Q

Find the partial fraction decomposition for:
3x-16
——————–
(x+4)(x-3)

A

4 1
β€”β€”- - ——–
x+4 x-3

60
Q

Find the partial fraction decomposition for

(x-3)(x^2+3)

A

-8 4x+7
——– + β€”β€”β€”-
x-3 x^2+3

Trig/calc (1 decks)

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