Title: M4.1 Trigonometry Functions and Vectors Quiz
Course: Technical Math for Industry > Section: Module 4.1 Trigonometry Functions and Vectors

Q: Which statement is true?
- The side opposite angle C and adjacent to angle A is . (Incorrect. The side opposite an angle is the side that does not form part of the angle. The side adjacent to an angle is one of the two sides that form the angle. is the hypotenuse and is not referred to as opposite or adjacent to an angle.)
- The side opposite angle C and adjacent to angle A is . (Incorrect. The side opposite an angle is the side that does not form part of the angle. The side adjacent to an angle is one of the two sides that form the angle.)
- The side opposite angle A and adjacent to angle C is . (Correct. The side opposite an angle is the side that does not form part of the angle. The side adjacent to an angle is one of the two sides that form the angle.)
- The side opposite angle A and adjacent to angle C is . (Incorrect. The side opposite an angle is the side that does not form part of the angle. The side adjacent to an angle is one of the two sides that form the angle.)
A: The side opposite angle A and adjacent to angle C is .

Q: A right triangle has acute angles and . If and , what are and ?
- and (Incorrect. The sine is correct, but it looks like you found instead of . C and D are complementary angles, so their cofunctions are equal: and)
- and (Incorrect. The cotangent is correct, but it looks like you found instead of . C and D are complementary angles, so their cofunctions are equal: and)
- and (Incorrect. It looks like you found and . C and D are complementary angles, so their cofunctions are equal: and)
- and (Correct. C and D are complementary angles, so their cofunctions are equal.)
A: and

Q: Which of the following could be trigonometric functions of the same angle?
- , , (Incorrect. Cotangent and tangent are reciprocals, as they should be. However, you cannot have this value of cosine for the same angle. Since , you could have and . If , then . But the lengths 1, 5, and 6 do not satisfy the Pythagorean Theorem.)
- , , (Incorrect. The sine and tangent could be for the same angle, but the cotangent is not the reciprocal of the tangent.)
- , , (Incorrect. The cosine and cotangent could be for the same angle, but the secant is not the reciprocal of the cosine.)
- , , (Correct. The sine and cosecant are reciprocals, as they should be. You can draw a right triangle with side opposite angle Y having length 9 and hypotenuse having length 41. Using the Pythagorean Theorem, you will find that the side adjacent to angle Y is 40. It follows that:)
A: , ,

Q: What is the value of to the nearest thousandth?
- 0.176 (Correct. First use your calculator to find . Then find the reciprocal of this: Finally, round off to 0.176.)
- 0.111 (Incorrect. You had your calculator set on radian mode. Make sure that it is first set in degree mode. Then use your calculator to find . After that, find the reciprocal of this number and round off.)
- 5.671 (Incorrect. You found . You need to take the reciprocal of this and round off.)
- 0.174 (Incorrect. You found . Use your calculator to find . Then find the reciprocal of this and round off.)
A: 0.176

Q: A ramp goes from a doorway of a building to the ground. The end of the ramp connected to the doorway is 1 foot above the ground. The horizontal distance from the bottom of the ramp to the building is 10 feet. What is the angle of elevation of the ramp to the nearest degree?
A: (a choice with no text)
Feedback: Correct. Call the angle of elevation A. The ramp is the hypotenuse of a right triangle with acute angle A. The length of the side opposite angle A equals the height above the ground of the other end of the ramp, so . The length of the side adjacent to angle A equals the horizontal distance to the building, so . This means that: So:
Feedback: Incorrect. You may have done the correct computation, finding , but then rounded off incorrectly.
Feedback: Incorrect. You did not have your calculator set in degree mode. Make sure you do this first before doing any other calculations.
Feedback: Incorrect. You found instead of finding .

Q: If angle X is an acute angle with , what is the value of ?
A: (a choice with no text)
Feedback: Incorrect. It appears that you correctly used the Pythagorean Theorem to find the length of the hypotenuse. However, you have found the value of . Remember that:
Feedback: Incorrect. You may have correctly used the definition of secant, but you did not use the Pythagorean Theorem. You can draw a triangle with and . Then use the Pythagorean Theorem to find the length h, of the hypotenuse: Solve this equation for h. Then use the definition of secant.
Feedback: Incorrect. It appears that you correctly used the Pythagorean Theorem to find the length of the hypotenuse. However, you have found the value of . Remember that:
Feedback: Correct. You can draw a triangle with and . Then use the Pythagorean Theorem to find the length h, of the hypotenuse: Now use the definition of secant:

Q: What is the value of x to the nearest hundredth?
- 18.55 (Incorrect. You probably set up the correct equation, , and solved it correctly. However, your calculator was not set to degrees.)
- 27.37 (Incorrect. You probably set up the equation incorrectly, setting equal to . A correct way to set up the equation is . Solve this equation for x and round off.)
- 13.35 (Correct. One way to set up a correct equation is to use the definition of cosine: Solve this equation for x: This rounds to 13.35.)
- 10.79 (Incorrect. You probably set up the equation incorrectly, setting equal to . A correct way to set up the equation is . Solve this equation for x and round off.)
A: 13.35

Q: If , what is the value of ?
- 1 (Incorrect. You may have mistakenly used a triangle and found . Refer to the triangle. From there you can see that . It follows that .)
- 2 (Incorrect. Refer to the triangle. From there you can see that . It follows that . You may have mistakenly found . You need to find .)
A: (a choice with no text)
Feedback: Correct. Refer to the triangle. From there you can see that . It follows that . So .
Feedback: Incorrect. Refer to the triangle. From there you can see that . It follows that . You may have mistakenly found .

Q: A 10-foot pole is supporting a tent and has a rope attached to the top. The rope is pulled straight and the other end is attached to a peg one foot above the ground. The rope and the pole form an angle that measures , as shown below.

Which expression shows the length of the rope?
A: (a choice with no text)
Feedback: Correct. Let x represent the length of the rope. If you draw a horizontal line from the top of the peg to the pole, you will form a right triangle. The length of the vertical leg of this triangle is feet. This is the side adjacent to the angle. The rope is the hypotenuse of the right triangle. So you can use cosine to set up your equation:
Feedback: Incorrect. It appears that you correctly knew to use cosine to set up an equation, but you made a mistake in figuring the role of the 1 foot peg. If you draw a horizontal line from the top of the peg to the pole, you will form a right triangle. The length of the vertical leg of this triangle is feet. This is the side adjacent to the angle. The length of the rope is the hypotenuse of this right triangle.
Feedback: Incorrect. It appears that you correctly knew to use cosine to set up an equation. You may have made a mistake in setting up the diagram first. If you draw a horizontal line from the top of the peg to the pole, you will form a right triangle. The length of the vertical leg of this triangle is feet. This is the side adjacent to the angle.
Feedback: Incorrect. It appears that you correctly knew to use cosine to set up an equation. You may have made a mistake in setting up the diagram first. If you draw a horizontal line from the top of the peg to the pole, you will form a right triangle. The length of the vertical leg of this triangle is feet. This is the side adjacent to the angle.

Q: What is the reference angle for ?
A: (a choice with no text)
Feedback: Incorrect. You may have correctly drawn the angle in standard position, starting on the x-axis, moving clockwise, and ending in second quadrant. However, the reference angle is the smaller angle formed by the terminal side and the x-axis. It has a value anywhere from to . You chose the larger angle formed by the terminal side and the x-axis.
Feedback: Correct. Draw the angle in standard position: it has its initial side on the x-axis and moves clockwise to the terminal side. The terminal side is in the second quadrant, because the angle measure is between (the negative x-axis) and (the positive y-axis). In particular: This tells you that the terminal side is above the negative x-axis. To say this another way, the x-axis and the terminal side form a angle. This is the reference angle.
Feedback: Incorrect. You may have correctly drawn the angle in standard position, starting on the x-axis, moving clockwise, and ending in second quadrant. However, a reference angle will have a value anywhere from to . It will never be negative.
Feedback: Incorrect. You may have correctly drawn the angle in standard position, starting on the x-axis, moving clockwise, and ending in second quadrant. However, the reference angle is the angle formed by the terminal side and the x-axis. You chose the angle formed by the terminal side and the y-axis.

Q: The angle is shown below in standard position, together with a unit circle.

Use the coordinates of the point of intersection of the terminal side and the circle to compute .
A: (a choice with no text)
Feedback: Incorrect. You may have confused cotangent with cosecant. This value is equal to . To calculate cotangent, use:
Feedback: Incorrect. You may have confused cotangent with cosine. This value is equal to . To calculate cotangent, use: .
Feedback: Incorrect. You may have confused cotangent with tangent. This value is equal to . To calculate cotangent, use:
Feedback: Correct. For an angle in standard position, with the terminal side intersecting the unit circle at the point , the value of the cotangent function is given by . Therefore:

Q: Compute .
- 2 (Incorrect. You may have correctly determined that the reference angle is and then confused secant with cosecant. To find the correct value of you must find the value of and then determine the correct sign. To determine the sign, use the signs of x and y in the fourth quadrant, remembering that:)
A: (a choice with no text)
Feedback: Correct. Draw in standard position. The terminal side is in the fourth quadrant and the reference angle is . Use the triangle to find the value of this function at the reference angle: Use the signs of x and y in the fourth quadrant to determine the sign of secant: So:
Feedback: Incorrect. You may have correctly determined that the reference angle is and then confused secant with cosecant. To find the correct value of you must find the value of and then determine the correct sign. To determine the correct sign, use the signs of x and y in the fourth quadrant, remembering that:
Feedback: Incorrect. You may have correctly determined that the reference angle is and found , but you made a mistake with the sign. Use the signs of x and y in the fourth quadrant to determine the sign of secant, remembering that:

Q: What signs are and ?
- They are both positive. (Incorrect. You may have been looking in the wrong quadrant. The terminal side of lies in Quadrant III.)
- and (Correct. The terminal side of lies in Quadrant III where x and y are both negative, so is negative and is positive.)
- and (Incorrect. You may have been looking in the wrong quadrant. The terminal side of lies in Quadrant III.)
- They are both negative. (Incorrect. Perhaps you thought that because the angle is negative, the values of these functions would be negative. The values depend on the quadrant in which the angle lies. The terminal side of lies in Quadrant III.)
A: and

Q: A circle has a diameter of 16 centimeters. Central angle FOG is drawn, determining an arc FG. The radian measure of angle FOG is . What is the length of arc FG in centimeters?
- cm (Incorrect. You may have used the number 8 cm as the value of s instead of the value of r. You need to use the formula for arc length with cm, and .)
- 12 cm (Incorrect. You probably used 16 cm as the value for r, but this is the diameter. You need to use 8 cm as the value of r and to find the value of s.)
- 6 cm (Correct. Use the formula for arc length. In this circle, cm, and , so .)
- cm (Incorrect. You probably used 16 cm as the value for s, but this is the diameter. Half this value, or 8 cm, is the value of r. Use that together with to find the value of s.)
A: 6 cm

Q: What radian measure is equivalent to ?
- radians (Correct. To convert from degrees to radians, multiply the degree measure by :)
- radians (Incorrect. It appears that you divided by . To convert from degrees to radians, multiply the degree measure by .)
- radians (Incorrect. It appears that you multiplied by . To convert from degrees to radians, multiply the degree measure by .)
- radians (Incorrect. You may have correctly found , but then reduced incorrectly. To convert from degrees to radians, multiply the degree measure by . Reduce the fraction using common factors of the numerator and denominator.)
A: radians

Q: What is the equivalent degree measure of radians written in simplest terms?
A: (a choice with no text)
Feedback: Incorrect. You mistakenly converted radians to degrees instead of radians.
Feedback: Incorrect. You multiplied by instead of . Also, when you write an angle in degrees, you must use a degree symbol, and this answer choice does not have one.
Feedback: Correct. To convert from radians to degrees, multiply the radian measure by and simplify:
Feedback: Incorrect. You probably correctly multiplied by , but you forgot to simplify.

Q: What is the value of ?
A: (a choice with no text)
Feedback: Incorrect. You may have correctly converted the angle to and found that it intersects the unit circle at the point . However, the value of cosine is the x-coordinate.
Feedback: Incorrect. It appears that you found the value of . Perhaps you made a mistake when converting from radians to degrees or finding the reference angle. You need to multiply a radian measure by to convert to degrees. So radians equals and the reference angle is . Find the x-coordinate of the point where intersects the unit circle.
Feedback: Incorrect. It appears that you found the value of . Convert to degrees: this equals . Find the x-coordinate of the point where intersects the unit circle.
Feedback: Correct. Convert the angle to degrees: This is in the fourth quadrant and its reference angle is . It intersects the unit circle at the point . The value of cosine is the x-coordinate:

Q: At how many points is on the interval ?
- 8 points (Incorrect. You may have correctly realized that each hill has two points where , but perhaps thought that each valley also had two such points. All of the points in each valley are below the horizontal axis, so the function has only negative values there.)
- 2 points (Incorrect. Perhaps you correctly recognized that there are two hills in the interval , but thought that there was a single point on each hill where , but this is not so. Notice that on the hill the value of goes up from 0 to 1, and then down from 1 to 0.)
- 4 points (Correct. On every hill in the graph, the value of first goes up from 0 to 1. At some point in between, it has the value . Then on the other side of the hill, the value of goes down from 1 to 0. At some point in between, it again has the value . Because the sine function equals at two points on each hill, and because there are two hills in the interval , there are a total of 4 points where .)
- 1 point (Incorrect. You may have been thinking about the fact that at exactly one point. However, the question is asking about the value of , not the value of . The first thing that you need to do is to look at each hill and recognize that the value of goes up from 0 to 1, and then down from 1 to 0.)
A: 4 points

Q: At how many points in the interval is ?
- 5 points (Incorrect. You may have been counting the number of high and low points, or points where the value of the function is 1 or . You need to count just the number of points where the function equals 1.)
- 1 point (Incorrect. The maximum value of the cosine function is 1, and it occurs at the top of each hill in the graph. Perhaps you were looking at only the point , but there are other points where the function equals 1.)
- 4 points (Incorrect. For any , there are 4 values of in this interval that yield . But this is not true for or . The graph is symmetric about the y-axis, and the value of cosine is 1 when . You need to find the other points where the value of cosine is 1.)
- 3 points (Correct. The maximum value of the cosine function is 1, and it occurs at the top of each hill in the graph. In the interval , there are three hilltops: at and .)
A: 3 points

Q: Which is not a correct description of the graph below?
- The graph of shifted to the left by units. (Correct. The graph shown is the graph of . This graph has a pattern that repeats every units, so if you were to shift it to the right or left by units you would get back the same graph. However, if you were to shift it to the left by units, the new graph would have a valley with the y-axis running through it. This would not match the graph shown.)
- The graph of shifted to the right by units. (Incorrect. If you shift the graph of by units to the right, this will result in a repeating valley and hill pattern, with a hill just to the right of the y-axis, and a valley just to the left. This matches the given graph.)
A: The graph of shifted to the left by units.
Feedback: Incorrect. The graph of is the same as the graph of shifted by units to the right. This will result in a repeating valley and hill pattern, with a hill just to the right of the y-axis, and a valley just to the left. This matches the given graph.
Feedback: Incorrect. The graph of is the same as the graph of shifted units to the right. But because the graph of has a pattern that repeats every units, this means that the graph of is exactly the same as the graph of . This is the graph shown.

Q: What are the amplitude and period of ?
- The amplitude is 10 and the period is . (Incorrect. The period is correct, but the amplitude is not. You may have thought that the amplitude was the height of the whole curve, but it is just the height of one hill. Another way to say this is that the amplitude of is given by . The period equals .)
- The amplitude is  5 and the period is . (Incorrect. The amplitude is correct, but the period is not. You may have thought that , or you may have made a mistake dividing by a fraction. Note that , so . The amplitude of is given by , and the period equals .)
- The amplitude is  5 and the period is . (Correct. The amplitude of is given by , so the amplitude of is 5 . The period equals . Now , so and the period equals:)
- The amplitude is 10 and the period is . (Incorrect. You may have thought that the amplitude was the height of the whole curve, but it is just the height of one hill. Another way to say this is that the amplitude of is given by . You may have thought that , or you may have made a mistake dividing by a fraction. Note that , so . The period of equals .)
A: The amplitude is  5 and the period is .

Q: The graph of a sine function is drawn. One full cycle goes from to and the low point on that cycle is . Which of the following functions could have this graph?
A: (a choice with no text)
Feedback: Correct. Because the low point is , the depth of the valley is 3. This is the amplitude of the function that was graphed. The amplitude of is also 3. The point satisfies . The length of one cycle is . This is equal to the period of the function that was graphed. The period of is also . So this could be the function that was graphed.
Feedback: Incorrect. This function has the same amplitude and period as the graphed function. However, its graph does not pass through the point . When , , not .
Feedback: Incorrect. This function and the graphed function both have 3 as the amplitude. However, they have different periods. The graphed function has a period of , while has a period of .
Feedback: Incorrect. You may have confused the effects of a and b in . The function has an amplitude of 5 and a period of . The graphed function has an amplitude of 3 and a period of .

Q: The graph of a cosine function is drawn. One full cycle goes from to and the low point on that cycle is . Which of the following functions could have this graph?
A: (a choice with no text)
Feedback: Correct. Because the low point is , the depth of the valley is 5. This is the amplitude of the function that was graphed. The amplitude of is also 5. The length of one cycle is 1. This is equal to the period of the function that was graphed. Now has a period equal to . So this could be the function that was graphed.
Feedback: Incorrect. Both this function and the graph have an amplitude of 5. However, the period of this function is , while the period of the graph is 1. Also, the graph of this function does not pass through the point .
Feedback: Incorrect. This function has the correct amplitude and period. However, when , the value of . So it does not pass through .
Feedback: Incorrect. Both this function and the graph have an amplitude of 5. However, the period of this function is , while the period of the graph is 1. Also, the graph of this function does not pass through the point .

Q: Which of the following graphs represents ?
A: (a choice with no text)
Feedback: Incorrect. This graph does have the symmetric shape of a cosine function and it has the correct amplitude. However, it does not have the correct period. Remember that the period of is given by .
Feedback: Incorrect. This graph has the symmetric shape of a cosine function and it does have the correct amplitude and period. However the orientation is incorrect. Because the coefficient is positive, this graph should have the same orientation as .
Feedback: Incorrect. This has the correct amplitude and period, but it is the graph of a sine function. Remember that the graph of a cosine function does not pass through the origin, and is symmetric.
Feedback: Correct. First, this graph is symmetric and has the shape of a cosine function. Second, because in the equation, the amplitude is . Finally, because , the period of this function is . This graph has one full cycle on the interval , so it has the correct period also.