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Trigonometry

Trigonometry Practice Test: Practice Test 4

Practice Test 4 for Trigonometry: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

For the complex number , find the modulus and the angle . Then, express this number in polar form .

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Question 1

For the complex number , find the modulus and the angle . Then, express this number in polar form .

  2. (correct answer)

Explanation: This problem has given us formulas, so we just need to plug in and and solve.

Question 2

Which of the following is not a solution to the following equation?

  1. (correct answer)
  5. ...

Explanation: We can factor the original expression as follows: So from this equation we conclude either that: or So any number that is not some integer multiple of away from these two solutions is not a solution to the original equation. The only such choice is , which is ; n is not an integer, therefore it is not a solution.

Question 3

Simplify your answer.

Convert to radians:

  1. (correct answer)

Explanation: We know that: Radians since the giving angle was in degrees then we multiply

Question 4

What is the domain of f(x) = sin x?

  1. All positive numbers and 0
  2. All negative numbers and 0
  3. All real numbers except 0
  4. All real numbers (correct answer)

Explanation: The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.

Question 5

Select the answer that correctly matches the following air navigation terms to their definitions.

  1. The heading of an airplane is the direction in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees. The groundspeed is the speed of the airplane in still air. The course of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The airspeed is the speed of the airplane relative to the ground. The drift angle is the positive difference between the heading and the course.
  2. The heading of an airplane is the direction in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees. The airspeed is the speed of the airplane in still air. The course of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The groundspeed is the speed of the airplane relative to the ground. The drift angle is the positive difference between the heading and the course. (correct answer)
  3. The course of an airplane is the direction in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees. The airspeed is the speed of the airplane in still air. The heading of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The groundspeed is the speed of the airplane relative to the ground. The drift angle is the positive difference between the heading and the course.
  4. The course of an airplane is the direction in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees. The groundspeed is the speed of the airplane in still air. The heading of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The airspeed is the speed of the airplane relative to the ground. The drift angle is the positive difference between the heading and the course.

Explanation: The heading of an airplane is the direction in which the airplane is pointed. The heading is measured clockwise from the north and expressed in degrees. The airspeed is the speed of the airplane in still air. The course of an airplane is the direction in which it moves relative to the ground. The course is measured clockwise from the north. The groundspeed is the speed of the airplane relative to the ground. The drift angle is the positive difference between the heading and the course. You may use vectors to represent airspeed and heading, direction and speed of wind, or groundspeed and course. The groundspeed vector is the resultant of the airspeed vector and the wind vector.

Question 6

Simplify using De Moivre's Theorem:

  1. (correct answer)

Explanation: We can use DeMoivre's formula which states: Now plugging in our values of and we get the desired result.

Question 7

Name the real part of this expression and the imaginary part of this expression: .

  1. Real: Imaginary:
  2. Real: Imaginary: (correct answer)
  3. Real: Imaginary:
  4. All parts are real; there are no imaginary parts
  5. All parts are imaginary; there are no real parts

Explanation: The real part of this expression includes any terms that do not have attached to them. Therefore the real part of this expression is 3. The imaginary part of this expression includes any terms with that cannot be further reduced; the imaginary part of this expression is .

Question 8

True or false:

.

  1. True
  2. False (correct answer)
  3. Cannot be determined

Explanation: The sum of sines is given by the formula .

Question 9

If , give the value of .

  1. (correct answer)

Explanation: Use the identity . Now we should divide both sides by : We can use the identity . or

Question 10

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 11

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 12

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 13

Find the value of

20

  1. (correct answer)

Explanation: Solving this problem begins with realizing that all three of our triangles are not only right triangles but isosceles and are therefore 45-45-90 triangles. That means in each triangle to get from the length of a leg to the length of the hypotenuse, we simply multiply by . Therefore, the hypotenuse of our bottom triangle is However, the hypotenuse of the bottom triangle is also the leg of the middle triangle. To find the hypotenuse of this triangle, we simply repeat the process. However, again the hypotenuse of the middle triangle is also the leg of the upper triangle. To find , the hypotenuse of the upper triangle, we simply repeat the process one last time.

Question 14

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 15

Which of the following shifts are incorrect?

  1. (correct answer)

Explanation: The actual shift for is .

Question 16

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 17

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 18

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 19

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 20

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 21

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 22

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 23

Which of the following describes the ratio of sine?

  1. (correct answer)

Explanation: Sine is by definition of sides in a right triangle is opposite side over hypotenuse. To remember this, use SOH CAH TOA. SOH: Sine=Opposite/Hypotenuse CAH: Cosine=Adjacent/Hypotenuse TOA: Tangent=Opposite/Adjacent

Question 24

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 .

Question 25

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 .