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Trigonometry

Trigonometry Practice Test: Practice Test 3

Practice Test 3 for Trigonometry: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A right triangle has side lengths of , , and . A similar right triangle has sides of , , and . What is ?

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Question 1

A right triangle has side lengths of , , and . A similar right triangle has sides of , , and . What is ?

  1. (correct answer)
  5. There is not enough information to determine.

Explanation: Similar triangles by defnition have proportional sides. We can divide corresponding parts in this case to find the scale factor. Corresponding parts are the two smallest sides, the medium sides, and the largest sides. Thus: is the scale factor. Then, we use this to find the missing side. Therefore, .

Question 2

Which of the given functions has the greatest amplitude?

  1. (correct answer)

Explanation: The amplitude of a function is the amount by which the graph of the function travels above and below its midline. When graphing a sine function, the value of the amplitude is equivalent to the value of the coefficient of the sine. Similarly, the coefficient associated with the x-value is related to the function's period. The largest coefficient associated with the sine in the provided functions is 2; therefore the correct answer is . The amplitude is dictated by the coefficient of the trigonometric function. In this case, all of the other functions have a coefficient of one or one-half.

Question 3

Triangle is equilateral with a side length of .

What is the height of the triangle?

  1. (correct answer)

Explanation: An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length, so..

Question 4

Solve for :

  1. (correct answer)

Explanation: First solve using the quadratic formula: This gives two potential solutions: The only value for where sine is 1 is . Using a calculator, we get Adding that to 360 givesus the angle's positive value, That's just one instance where the sine is -0.75. We also need to find the other angle below the x-axis by adding . So our three values for theta are

Question 5

Which of the following is the graph of the inverse of with ?

  1. Screen shot 2020 08 27 at 10.45.10 am
  2. Screen shot 2020 08 27 at 10.46.15 am (correct answer)
  3. Screen shot 2020 08 27 at 10.47.13 am
  4. Screen shot 2020 08 27 at 10.47.06 am

Explanation: Note that the inverse of is not , that is the reciprocal. The inverse of is also written as . The graph of with is as follows. Screen shot 2020 08 27 at 10.45.10 am And so the inverse of this graph must be the following with and Screen shot 2020 08 27 at 10.46.15 am

Question 6

Rt_triangle_letters

In this figure, if angle , side , and side , what is the measure of angle ?

  1. (correct answer)
  3. Undefined

Explanation: Since , we know we are working with a right triangle. That means that . In this problem, that would be: Plug in our given values:

Question 7

A plank has one end on the ground and one end off the ground. What is the measure of the angle formed by the plank and the ground?

  1. (correct answer)

Explanation: The length of the plank becomes the hypotenuse of the triangle, while the distance between the plank and the ground becomes the length of one side. To solve for the angle between the plank and the ground, you must find the value of . The sine of the angle is the value of the opposite side over the hypotenuse, which are values that we know.

Question 8

Identify the phase shift of the following equation.

  1. (correct answer)

Explanation: If we use the standard form of a sine function the phase shift can be calculated by . Therefore, in our case, our phase shift is

Question 9

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

1.

  1. (correct answer)

Explanation: Using the quotient identities for trig functions, you can rewrite, and Then the fraction becomes

Question 10

Find the exact value of the expression:

  1. (correct answer)
  2. The expression is undefined.

Explanation: There are two ways to solve this problem. If one recognizes the identity , the answer is as simple as: If one misses the identity, or wishes to be more thorough, you can simplify:

Question 11

If c=70, a=50, and find to the nearest degree.

  3. and
  4. and
  5. no solution (correct answer)

Explanation: Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From , we get . In this equation, if , no that satisfies the triangle can be found. If and there is a right triangle determined. Finally, if , two measures of can be calculated: an acute and an obtuse angle . In this case, there may be one or two triangles determined. If , then the is not a solution. In this problem, , which means that there are no solutions to that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

Question 12

Solve the following trigonometric equation:

for

  1. (correct answer)
  4. The equation does not have a solution.

Explanation: Since can be written as: . We can't have . Therefore . This means that where k is an integer. since . We have x=0 is the only number that satisfies this property.

Question 13

Change angle to degrees.

  1. (correct answer)

Explanation: In order to change an angle into degrees, you must multiply the radian by . Therefore, to solve:

Question 14

Which of the following trigonometric identities is INCORRECT?

  1. (correct answer)

Explanation: Cosine and sine are not reciprocal functions. and

Question 15

If , , and find to the nearest degree.

  1. (correct answer)

Explanation: The problem gives the lengths of three sides and asks to find an angle. We can use the Law of Cosines to solve for the angle. Because we are solving for , we use the equation: Substituting the values from the problem gives Isolating by itself gives

Question 16

Determine the quadrant that contains the terminal side of an angle .

  1. (correct answer)

Explanation: Each quadrant represents a change in degrees. Therefore, an angle of radians would pass through quadrants , , , and end in quadrant . The movement of the angle is in the clockwise direction because it is negative.

Question 17

In triangle , , and . To the nearest tenth, what is ?

  1. (correct answer)

Explanation: By the Law of Cosines, or, equivalently, Substitute:

Question 18

Find the area of given that inches, inches, and .

  1. (correct answer)

Explanation: Using the formula for area of a triangle equal to , drawing and labelling its sides, angles, and height h, then using triangle trigonometry and substitution, we can derive the formulae , where R is equal to area. This can be used to find the area of a triangle when we know two of its sides and the included angle. Plugging in, we get: Therefore the area of this triangle is 129.2 square inches.

Question 19

Which of the following is the correct definition of a phase shift?

  1. A measure of the length of a function between vertical asymptotes
  2. The distance a function is shifted diagonally from the general position
  3. The distance a function is shifted horizontally from the general position (correct answer)
  4. The distance a function is shifted vertically from the general position

Explanation: Take the function for example. The graph for is If we were to change the function to , our phase shift is . This means we need to shift our entire graph units to the left. Our new graph is the following

Question 20

Given the accompanying right triangle where and , determine the measure of to the nearest degree.

Right_triangle

  1. (correct answer)

Explanation: We are given two sides of the right triangle, namely the hypotenuse and the opposite side of the angle. Therefore, we simply use the sine function to determine the angle:
 In order to isolate the angle we must apply the inverse sine function to both sides:

Question 21

Convert radians into degrees.

  1. (correct answer)

Explanation: Recall the definition of "radians" derived from the unit circle: The quantity of radians given in the problem is . All that is required to convert this measure into degrees is to denote the unknown angle measure in degrees by and set up a proportion equation using the aforementioned definition relating radians to degrees: Cross-multiply the denominators in these fractions to obtain: or . Canceling like terms in these equations yields Hence, the correct angle measure of in degrees is .

Question 22

In the figure below, is a diagonal of quadrilateral . has a length of . is congruent to .

Screen shot 2020 08 27 at 4.39.20 pm

Which of the following is a true statement?

  1. The area of quadrilateral is . (correct answer)
  2. The area of quadrilateral is .
  3. The perimeter of quadrilateral is .
  4. The perimeter of quadrilateral is .

Explanation: Since and are perpendicular, is a right angle. Since no triangle can have more than one right angle, and is isosceles, must be congruent to . Since angle CBD is congruent to and measures 90 degrees, and can be calculated as follows: Therefore, and are both equal to 45 degrees. is a 45-45-90 triangle. Therefore, the ratio between side lengths and hypotenuse is . Anyone of the four side lengths of quadrilateral must, therefore, be equal to . To find the area of , multiply two side lengths: .

Question 23

Which is true of the relationship between the arc measure and the central angle as shown below?

Screen shot 2020 08 27 at 4.09.48 pm

  1. They are equal (correct answer)
  2. The central angle is half of the arc length
  3. The central angle will always be a right angle
  4. The arc length is half of the central angle

Explanation: Every arc has a measure that is equal to the measure of the central angle that creates the arc. This is because the measure of the angle determines the distance around the circumference that the arc makes.

Question 24

Given a right triangle where , find the missing side.

  1. (correct answer)

Explanation: Since the triangle in question is a right triangle we can use the Pythagorean Theorem. First, we must determine which sides we are given. Since the function we are given is cosine, we know that we are given the adjacent side and hypotenuse. Therefore, setting up the equation: Where, and are given. Solving the above equation: We toss out the negative solution since the length of a side must be positive.

Question 25

Figure1

Given sides , and angle determine the corresponding value for

  1. (correct answer)
  3. Undefined

Explanation: The Law of Sines is used here since we have Side - Angle - Side. We setup our equation as follows: Next, we substitute the known values: Now we cross multiply: Divide by 10 on both sides: Finally taking the inverse sine to obtain the desired angle: