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Trigonometry

Trigonometry Practice Test: Practice Test 1

Practice Test 1 for Trigonometry: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Change angle to degrees.

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Question 1

Change angle to degrees.

  1. (correct answer)

Explanation: In order to change radians to degrees, we need to multiply the radian agle measure by .

Question 2

Find the zeros of the above equation in the interval

.

  1. (correct answer)

Explanation: Therefore, and that only happens once in the given interval, at , or 45 degrees.

Question 3

A triangle has three angles , and such that and . The side opposite to measures units in length. How long is the side opposite of ?

  1. (correct answer)

Explanation: A triangle with a angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units. Therefore, to make the above statement true .

Question 4

For which values of , where in the unit circle, is undefined?

  1. (correct answer)

Explanation: Recall that . Since the ratio of any two real numbers is undefined when the denominator is equal to , must be undefined for those values of where . Restricting our attention to those values of between and , when or . Hence, is undefined when or .

Question 5

True or False: The inverse of the function is also a function.

  1. True
  2. False (correct answer)

Explanation: Consider the graph of the function . It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that is a function. Screen shot 2020 08 27 at 11.59.46 am Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point. Screen shot 2020 08 27 at 12.00.18 pm This is not true, and we can also see that if we graph the inverse of () that this does not pass the vertical line test and therefore is not a function. If you wish to graph the inverse of , then you must restrict the domain so that your graph will pass the vertical line test. Screen shot 2020 08 27 at 12.00.40 pm

Question 6

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 7

Use De Moivre's Theorem to evaluate .

  4. (correct answer)

Explanation: First convert this point to polar form: Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is We are evaluating Using DeMoivre's Theorem: DeMoivre's Theorem is We apply it to our situation to get: which is coterminal with since it is an odd multiplie

Question 8

Simplify the following expression using trigonometric identities:

  1. (correct answer)
  5. Can not be further reduced

Explanation: In order to simplify the given equation we should first try to determine if the Pythagorean Theorem as applicable to trigonomety can be utilized. We do this first due to the higher degree of the functions involved. We can notice that if we group the higher order sine and the higher order cosine, that we can in fact pull out some common terms: Now we notice that we can further group the terms: The first term in the previous equation is in fact the Pythagorean Theorem as applied to trigonometry and the second term is the sum of two angles with respect to the sine function: This reduced simply to the sum function for sine:

Question 9

Which of the following completes the identity

  2. (correct answer)

Explanation: This formula is able to be derived directly from the identities for the sum and difference of cosines, .

Question 10

Solve the following equation by squaring both sides:

  3. (correct answer)

Explanation: We begin with our original equation: (Pythagorean Identity) Looking at the unit circle we see that at and . We must plug these back into our original equation to validate them. Checking Checking And so our only solution is

Question 11

Using trigonometric identities determine whether the following is valid:

  1. False (correct answer)
  2. True
  3. Uncertain
  4. Only valid in the range of:
  5. Only valid in the range of:

Explanation: We can choose either side to work with to attempt to obtain the equivalency. Here we will work with the right side as it is the more complex. First, we want to eliminate the negative angles using the appropriate relations. Sine is odd and therefore, the negative sign comes out front. Cosine is even which is interpreted by dropping the negative out of the equation: The squaring of the sine in the denominator makes the sine term positive, i.e. The numerator is the double angle formula for sine: The denominator is recognized to be the pythagorean theorem as it applies to trigonometry: The final reduced equation is: Thus proving that the equivalence is false.

Question 12

Find the difference of the two vectors, which ends at and ending at .

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

Explanation: When finding the difference of two vectors, you must subtract the x and y components separately.

Question 13

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 14

Simplify .

  1. (correct answer)

Explanation: To solve this problem, make sure you set it up to multiply the entire parentheses by itself (a common mistake it to try to simply distribute the exponent 2 to each of the terms in the parentheses.) (recall that ) Please note that while the answer choice is not incorrect, it is not fully simplified and therefore not the correct choice.

Question 15

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 16

Which of the following is the graph of ?

  1. Screen shot 2020 08 27 at 3.40.13 pm
  2. Screen shot 2020 08 27 at 3.44.11 pm
  3. Screen shot 2020 08 27 at 3.42.14 pm (correct answer)
  4. Screen shot 2020 08 27 at 3.44.04 pm

Explanation: To derive the graph of , recall that . The graph of is Screen shot 2020 08 27 at 3.40.13 pm and the graph of is Screen shot 2020 08 27 at 3.40.18 pm Vertical asymptotes will occur in the graph of whenever . This is because the denominator of the tangent function will be equal to zero whenever the cosine function is equal to zero and then the entire function will be undefined at those points. Wherever cosine crosses the x-axis a vertical asymptote will occur. If we overlay the sine and cosine graphs we see the following: Screen shot 2020 08 27 at 3.40.24 pm So our tangent graph will follow the same form as the sine and cosine graphs when they are increasing, but will have vertical asymptotes wherever cosine crosses the x-axis. Screen shot 2020 08 27 at 3.42.06 pm And we are left with our graph of Screen shot 2020 08 27 at 3.42.14 pm

Question 17

True or False: All solutions found from squaring both sides of a trigonometric function are valid should be given as a final answer.

  1. True
  2. False (correct answer)

Explanation: This is not true. This is because when squaring both sides and then plugging back into the original equations, some of our solutions may be extraneous solutions. Therefore, when solving a trigonometric equation by squaring both sides, all solutions found must be plugged back into the original equation and validated.

Question 18

Using trigonometric identities prove whether the following is valid:

  1. True (correct answer)
  2. False
  3. Uncertain
  4. Only in the range of:
  5. Only in the range of:

Explanation: We can work with either side of the equation as we choose. We work with the right hand side of the equation since there is an obvious double angle here. We can factor the numerator to receive the following: Next we note the power reducing formula for sine so we can extract the necessary components as follows: The power reducing formula must be inverted giving: Now we can distribute and reduce: Finally recalling the basic identity for the cotangent: This proves the equivalence.

Question 19

Solve the following equation. Find all solutions such that .

  1. (correct answer)
  5. No Solution

Explanation: ; Divide both sides by 2 to get ; take the inverse sine on both sides ; the left side reduces to x, so At this point, either use a unit circle diagram or a calculator to find the value. Keep in mind that the problem asks for all solutions between and . If you use a calculator, you will only get as an answer. So we need to find another angle that satisfies the equation .
Problem_1_set_1

Question 20

If , and , determine the measure of to the nearest degree.

  1. (correct answer)

Explanation: This is a straightforward Law of Cosines problem since we are given three sides and desire one of the corresponding angles in the triangle. We write down the Law of Cosines to start: Substituting the given values: Isolating the angle: The final step is to take the inverse cosine of both sides:

Question 21

If the height of the stair is 2 ft, and the length of the stair is 3 ft, how long must the ramp be to cover the stair?

Screen_shot_2014-02-15_at_2.24.54_pm

  1. 3.6 ft (correct answer)
  2. 2.5 ft
  3. 13 ft
  4. 5 ft
  5. 10 ft

Explanation: Use the Pythagorean triangle to solve for the third side of the triangle. Simplify and you have the answer:

Question 22

Express the complex number in rectangular form.

  1. (correct answer)

Explanation: To convert this number to rectangular form, first think about what and are equal to. Because , we can use a 30-60-90o reference triangle in the 3rd quadrant to determine these values. D5g1377 Now plug these in and continue solving:

Question 23

Which of the following is positive?

  1. (correct answer)

Explanation: When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

Question 24

Solve the following system:

Screen shot 2020 05 21 at 10.30.36 am

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

Explanation: A number x is a solution if it satisfies both equations. We note first we can write the first equation in the form : We know that for all reals. This means that there is no x that satisifies the first inequality. This shows that the system cannot satisfy both equations since it does not satisfy one of them. This shows that our system does not have a solution.

Question 25

The function shown below has an amplitude of   and a period of  .

  1. (correct answer)

Explanation: The amplitude is always a positive number and is given by the number in front of the trigonometric function. In this case, the amplitude is 4. The period is given by , where b is the number in front of x. In this case, the period is .