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Trigonometry

Trigonometry Practice Test: Practice Test 5

Practice Test 5 for Trigonometry: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Simplify the following trigonometric function in fraction form:

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Question 1

Simplify the following trigonometric function in fraction form:

  1. (correct answer)

Explanation: To determine the value of the expression, you must know the following trigonometric values: Replacing these values, we get:

Question 2

Simplify the equation using identities:

  1. (correct answer)
  2. 1

Explanation: There are a couple valid strategies for solving this problem. The simplest is to first factor out from both sides. This leaves us with: Next, substitute with the known identity to get: From here, we can eliminate the quadratic by converting: giving us Thus,

Question 3

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 4

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:

Question 5

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 6

Simplify the expression:

  1. (correct answer)
  5. The equation cannot be further simplified.

Explanation: The expression represents a difference of squares. In this case, the product is (remember that 1 is also a perfect square). One Pythagoran identity for trigonometric functions is: Thus, we can say that the most simplified version of the expression is .

Question 7

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 8

Solve for :

  1. (correct answer)

Explanation: Subtracting 5 from both sides gives the quadratic equation Using the quadratic formula gives: The cosine cannot be 3 because that's greater than 1.

Question 9

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 10

Solve each quation over the interval

  1. (correct answer)

Explanation: Rearrange the equation so that, . Take the square of both sides and find the angles for which . These two angles are and .

Question 11

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 12

Trig_id

What is if and ?

  1. (correct answer)

Explanation: In order to find we need to utilize the given information in the problem. We are given the opposite and adjacent sides. We can then, by definition, find the of and its measure in degrees by utilizing the function. Now to find the measure of the angle using the function. If you calculated the angle's measure to be then your calculator was set to radians and needs to be set on degrees.

Question 13

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 14

Find if and .

  1. (correct answer)

Explanation: The double-angle identity for sine is written as and we know that Using , we see that , which gives us Since we know is between and , sin is negative, so . Thus, . Finally, substituting into our double-angle identity, we get

Question 15

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 16

Which of the following represents graphically?

  1. Screen shot 2020 08 04 at 12.40.28 pm
  2. Screen shot 2020 08 04 at 12.45.00 pm
  3. Screen shot 2020 08 04 at 12.38.00 pm (correct answer)
  4. Screen shot 2020 08 04 at 12.44.20 pm

Explanation: To represent complex numbers graphically, we treat the x-axis as the "axis of reals" and the y-axis as the "axis of imaginaries." To plot , we want to move 6 units on the x-axis and -3 units on the y-axis. We can plot the point P to represent , but we can also represent it by drawing a vector from the origin to point P. Both representations are in the diagram below. Screen shot 2020 08 04 at 12.38.00 pm

Question 17

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 18

Find all angles when .

  1. and
  2. and
  3. and (correct answer)
  4. and

Explanation: We can use reference angles, inverse trig, and a calculator to solve this problem. Below is a table of reference angles. Screen shot 2020 07 30 at 11.05.57 am We have so . Next, think about where sine is negative, or reference the Function Signs column of the above table. Sine is negative in Quadrants III and IV. In Quadrant III, . In Quadrant IV, . If this problem asked for values of between and , our work would be done, but this problem does not restrict the range, so we need to give all possible values of by generalizing our answers. To do this, we must understand that all angles that are coterminal to and will also be solutions. Coterminal angles add or subtract multiples of . To write this generally, we write: and .

Question 19

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 20

Find the value of in the triangle below.

16

  1. (correct answer)

Explanation: The first two things to recognize regarding our tirangle are 1) it is a right triangle and 2) it is an isosceles triangle. The two congruent sides tell us that the two non-right angles are also congruent, and a little quick math tells us that they each equal 45 degrees. This means our right triangle is not just any right triangle but a 45-45-90 triangle. This is important because the sides of every 45-45-90 triangle follow the same ratio. The two legs are obviously always congruent to each other (being isosceles), but to find the hypotenuse, we simply have to multiply the length of a leg by . Given this fact we would be in good shape if we had the length of a leg and needed the hypotenuse. But we have the hypotenuse and need the leg, which we means we need to work backwards going this way, we need to divide the length of the hypotenuse by . Therefore, However, general practice in mathematics doesn't allow us to leave a square root in the denominator. We solve this problem by rationalizing the denominator, which is accomplished by multiplying the numerator and the denominator by . This effectively eliminates the square root in the denominator and provides our answer.

Question 21

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 22

If , give in terms of .

  1. (correct answer)

Explanation: We need to use the identity .

Question 23

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 24

Knowing that the arc length of a sector and the angle measure is , what is the area of the sector?

  3. (correct answer)

Explanation: To solve this problem we must know the formula for finding the arc length of a sector. This formula is . With the given information we are able to solve for the radius which we can then use to solve for the area of the sector itself. Now we can plug this radius into the formula to solve for the area of a sector.

Question 25

What is the value of from the unit circle?

  1. (correct answer)

Explanation: From the unit circle, the value of . This can be found using the coordinate pair associated with the angle which is . Recall that the pair are .