Title: Unit 19 Quiz
Course: Technical Math for Industry

Q: Side is opposite which angle?
A: (a choice with no text)
If wrong: Incorrect. Side is part of the top right triangle, while is part of the bottom right triangle. The side opposite would be .
If right: Correct. The side opposite an angle is the side that does not form part of the angle. Side is part of the top right triangle and does not form part of .
If wrong: Incorrect. The side opposite an angle is the side that does not form part of the angle. Side is part of the top right triangle. It forms part of , so it is adjacent to this angle.
If wrong: Incorrect. Side is part of the top right triangle, while is part of the bottom right triangle. The side opposite would be .

Q: What are and ?
- and (Incorrect. The sine is correct, but it looks like you confused secant with cosine. Remember that secant is hypotenuse over adjacent.)
- and (Incorrect. The secant is correct, but it looks like you confused sine with cosecant. Remember that sine is opposite over hypotenuse.)
- and (Correct. The side opposite angle D has length 9, the side adjacent to angle D has length 40, and the hypotenuse has length 41.)
- and (Incorrect. It looks like you confused sine with cosecant and secant with cosine. Remember that sine is opposite over hypotenuse and secant is hypotenuse over adjacent.)
A: and

Q: Acute angle has and . Which are values of and ?
- and (Incorrect. The cosecant is the reciprocal of sine, as it should be. However, cosine should be the reciprocal of secant.)
- and (Incorrect. The cosine is the reciprocal of secant, as it should be. However, cosecant should be the reciprocal of sine.)
- and (Incorrect. Cosecant should be the reciprocal of sine, and cosine should be the reciprocal of secant.)
- and (Correct. The cosecant is the reciprocal of the sine and the cosine is the reciprocal of the secant.)
A: and

Q: What is the value of to the nearest thousandth?
- 0.629 (Incorrect. You found . Use your calculator to find . Then find the reciprocal of this and round off.)
- 1.287 (Correct. First use your calculator to find . Then find the reciprocal of this: Finally, round off to 1.287.)
- 1.589 (Incorrect. You found . Remember that secant is the reciprocal of cosine, not sine. Use your calculator to find . Then find the reciprocal of this and round off.)
- 3.750 (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.)
A: 1.287

Q: Use your calculator to find the angle, to the nearest hundredth of a degree, whose cosine value is .
A: (a choice with no text)
If wrong: Incorrect. You may have found , , or just . You need to find . You also may not have had your calculator set in degree mode.
If wrong: Incorrect. You found instead of finding . You also did not have your calculator set in degree mode.
If wrong: Incorrect. You did not have your calculator set in degree mode. Make sure you do this first before doing any other calculations.
If right: Correct. On a scientific calculator, first divide 1 by 7, then press the 2nd key and COS. Do this in the reverse order for a graphing calculator.

Q: What is the value of ?
A: (a choice with no text)
If wrong: Incorrect. You found the value of . First use the Pythagorean Theorem to find b: After solving for b, use the definition of cosecant:
If wrong: Incorrect. It appears that you found the value of . Remember that the two acute angles in a right triangle will have different trigonometric ratios. First use the Pythagorean Theorem to find b: After solving this equation, substitute the value of b for the opposite side in the definition of cosecant.
If right: Correct. The Pythagorean Theorem gives you: Then:
If wrong: Incorrect. You may have used the definition of cosecant, but you incorrectly used the Pythagorean Theorem to find b. The correct equation is: After solving this equation, substitute the value of b for the opposite side in the definition of cosecant.

Q: What is the value of y to the nearest hundredth?
- 6.29 (Correct. One way to set up a correct equation is to use the definition of sine: Solve this equation for y: This rounds to 6.29.)
- 9.64 (Incorrect. You probably set up the correct equation, , and solved it correctly. However, your calculator was not set to degrees.)
- 8.10 (Incorrect. You may have set up the incorrect equation, using tangent. The correct equation to use is . Solve this equation for y and round off.)
- 15.89 (Incorrect. You probably set up the ratio incorrectly, equating with . A correct way to set up the equation is . Solve this equation for y and round off.)
A: 6.29

Q: What is the value of p in the triangle below?
- 4 (Incorrect. Perhaps you referred to a triangle. However, you can use the fact that to conclude that this is a triangle. You can then use the definition of cosine to find the value of p.)
A: (a choice with no text)
If wrong: Incorrect. You may have figured out that this is a triangle, but then set up your equation incorrectly. One equation that you can use to find the value of p is:
If wrong: Incorrect. You may have figured out that this is a triangle and set up the correct equation: At this point, however, you solved the equation incorrectly.
If right: Correct. You can use the given sides to find: Refer to the triangle. From there you can see that . It follows that . Use the definition of cosine: This implies that:

Q: A rectangular park had a dirt path across its diagonal that was 100 yards long. The diagonal and the long side of the park formed an angle that measured .

A person walked along the sidewalks outside the park, from the start to the end of the path, as shown by the arrows.

Which expression shows the distance that he walked?
- yards (Correct. You can use cosine to find x and sine to find y: and Multiply both sides of both equations by 100: and Add the two distances and use a calculator to find a numerical value.)
- yards (Incorrect. You found the long side and multiplied by 2, but the two sides are not equal. You need to use cosine to find x and sine to find y: and Solve for x and y, add the results and use a calculator to get a numerical value.)
- yards (Incorrect. You may have thought that, because the lengths of the two sides are involved, and they are opposite and adjacent to the angle, that somehow cotangent was involved in the answer. However, cotangent is not useful in this problem, because it would give you a ratio with the two variables. Instead, you need to use cosine to find x and sine to find y: and Solve for x and y, add the results and use a calculator to get a numerical value.)
- yards (Incorrect. You may have thought that, because the lengths of the two sides are involved, and they are opposite and adjacent to the angle, that somehow tangent was involved in the answer. However, tangent is not useful in this problem, because it would give you a ratio with the two variables. Instead, you need to use cosine to find x and sine to find y: and Solve for x and y, add the results and use a calculator to get a numerical value.)
A: yards

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

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)
If wrong: Incorrect. You may have made a mistake while simplifying or you may have calculated the incorrect ratio. This value is equal to . To calculate cotangent, use:
If wrong: Incorrect. You may have confused cotangent with cosine. This value is equal to . To calculate cotangent, use:
If wrong: Incorrect. You may have made a mistake while simplifying or you may have calculated the incorrect ratio. This value is equal to . To calculate cotangent, use:
If right: 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: Determine .
- 1 (Incorrect. You may have correctly determined that the reference angle is and then confused cosine with cotangent. 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:)
A: (a choice with no text)
If right: 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 cosine: So:
If wrong: 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 cosine, remembering that:
If wrong: Incorrect. You may have correctly determined that the reference angle is and then confused cosine with cotangent. 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:

Q: What signs are and ?
- and (Correct. The terminal side of lies in Quadrant IV where x is positive and y is negative, so is positive and is negative.)
- They are both positive. (Incorrect. You may have been looking in the wrong quadrant. The terminal side of lies in Quadrant IV.)
- 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 IV.)
- and (Incorrect. You may have been looking in the wrong quadrant. The terminal side of lies in Quadrant IV.)
A: and

Q: The radius of a circle is 2 feet. Central angle AOB cuts off arc AB. The length of arc AB is yards. What is the radian measure of angle AOB?
A: (a choice with no text)
If wrong: Incorrect. You probably used the correct formula , with s representing arc length and r representing the radius, but you forgot to convert one of them. In order to use this formula you need to have the two measurements in the same units.
If wrong: Incorrect. You probably divided the radius by the arc length. You need to use the formula , dividing the arc length by the radius.
If wrong: Incorrect. You probably divided the radius by the arc length and you forgot to convert units. You need to use the formula , dividing the arc length by the radius. These two measurements must be in the same units.
If right: Correct. To use the formula for radian measure, you must first have the radius and the arc length in the same units, so you must convert one of them. Now , so:

Q: What radian measure is equivalent to ?
- radians (Incorrect. It appears that you multiplied by . To convert from degrees to radians, multiply the degree measure by .)
- radians (Correct. 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.)
- radians (Incorrect. It appears that you divided by . To convert from degrees to radians, multiply the degree measure by .)
A: radians

Q: What is the equivalent degree measure of radians written in simplest terms?
A: (a choice with no text)
If wrong: 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.
If wrong: Incorrect. You probably correctly multiplied by , but you forgot to simplify.
If wrong: Incorrect. You probably correctly multiplied by , but you left out the negative sign.
If right: Correct. To convert from radians to degrees, multiply the radian measure by and simplify:

Q: An angle in standard position intersects the unit circle at the point .

Which set of information about this angle does this point give you?
A: (a choice with no text)
If wrong: Incorrect. You may have correctly determined that the point is where the angle radians or intersects the unit circle. However, you forgot which coordinate corresponds to which function. The x-coordinate gives you the value of the cosine function at that angle, and the y-coordinate gives you the value of the sine function at that angle.
If right: Correct. The point is where the angle radians or intersects the unit circle. The x-coordinate gives you the value of the cosine function at that angle, and the y-coordinate gives you the value of the sine function at that angle. So:
If wrong: Incorrect. You probably worked with the point , which corresponds to the angle . However, the point is where the angle radians or intersects the unit circle.
If wrong: Incorrect. You probably worked with the point , which corresponds to the angle and gives you this information. However, the point is where the angle radians or intersects the unit circle.

Q: For which values of in the interval is the function equal to 0?
A: (a choice with no text)
If wrong: Incorrect. The sine function is equal to zero at these two values of , but they are not the only values on this interval for which the function equals 0. Remember that the graph of sine has a hill/valley pattern that repeats forever in both directions. Because of this, values of the function will repeat with regularity.
If right: Correct. You have seen that the function equals 0 at and . The graph of sine has a hill/valley pattern that repeats forever in both directions. Each hill and each valley is units wide and ends on the horizontal axis, or with a value of 0. So the function will continue to be equal to 0 at every multiple of . All the multiples of in the interval are .
If wrong: Incorrect. The sine function is equal to zero at these two values of , but they are not the only values on this interval for which the function equals 0. Remember that the graph of sine has a hill/valley pattern that repeats forever in both directions. Because of this, values of the function will repeat with regularity.
If wrong: Incorrect. The sine function is equal to zero at these two values of , but they are not the only values on this interval for which the function equals 0. Remember that the graph of sine has a hill/valley pattern that repeats forever in both directions. Because of this, values of the function will repeat with regularity.

Q: Which points are on the graph of ?
- and (Incorrect. These two points are on the graph of . The graph of cosine has a hill and valley pattern like sine, but the hills and valleys are not located in the same places, so they do not have the same points.)
- and (Incorrect. The first point is on the graph of cosine, but the second point is on the graph of sine. When : So the point is on the graph of cosine.)
- and (Correct. The graph of cosine has a repeating hill and valley pattern. The pattern repeats after intervals of length in both directions. In particular: So the point is on the graph of cosine. You also know that: So the point is also on the graph of cosine.)
- and (Incorrect. The second point is on the graph of cosine, but the first point is not. The graph of cosine has a repeating hill and valley pattern. The pattern repeats after intervals of length in both directions. In particular: So the point is on the graph of cosine.)
A: and

Q: Which equation is shown graphed below?
A: (a choice with no text)
If wrong: Incorrect. You may have thought that the effect of adding a constant to the input was to shift the graph of the original function to the right, but actually it will shift to the left. So the graph of will be the same as the graph of cosine, which is not the graph shown.
If right: Correct. The effect of subtracting from the input of the function is to shift the graph of to the right by units. So, for example, the valley that was on the interval in the graph of becomes a valley with the y-axis running through the center in the graph of . The hill and valley pattern repeats on both sides and this is a description of the graph shown.
If wrong: Incorrect. You may have recognized that the given graph is symmetric about the y-axis, as is the graph of . However, the given graph has a valley with the y-axis running through it, while the graph of has a hill with the y-axis running through it.
If wrong: Incorrect. You may have miscalculated the effect of the on the graph of cosine. It will shift the graph of to the left by units. So the graph of will have a hill just to the left of the y-axis, and a valley just to the right. This does not match the given graph.

Q: What are the amplitude and period of ?
- The amplitude is 3 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 . Perhaps you thought that the 3 in front of the x would make the period 3 times as long, but it does just the opposite. The period equals .)
- The amplitude is 3 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 and the period is . (Incorrect. The amplitude is correct, but the period is not. You may have thought that the 3 in front of the x would make the period 3 times as long, but it does just the opposite. The amplitude of is given by , and the period equals .)
- The amplitude is and the period is . (Correct. The amplitude of is given by , so the amplitude of is . The period equals .)
A: The amplitude is and the period is .

Q: Which of the following is the graph of ?
A: (a choice with no text)
If wrong: Incorrect. This has the right amplitude and period, but this is the graph of a cosine function. In particular, the graph of must pass through the point , while this graph does not.
If wrong: Incorrect. You probably made a mistake with the calculation of the period. The amplitude of this graph and is 2. However, this graph has half a cycle in the interval , so it has a period of . But has a period of .
If right: Correct. This is the graph of a sine function: it has the repeating hill and valley pattern and passes through the origin. The amplitude of this graph and is 2. This graph has 3 cycles in the interval , or a period of . The same is true for , so the graph and the function match.
If wrong: Incorrect. It looks like you confused the effect of a and b in . The amplitude of this graph is 3, while the amplitude of is 2. This graph has 2 cycles in the interval , or a period of . The period of is .

Q: Which of the following is the graph of ?
A: (a choice with no text)
If wrong: Incorrect. You probably made a mistake with the calculation of the period. The amplitude of this graph and is 3. However, this graph has a period of , while has a period of .
If right: Correct. This graph has the shape of a cosine function. The amplitude of this graph and is 3. This graph has one cycle on the interval , so it has a period of . The function has a period equal to . The effect of the negative sign is to reflect the graph around the y-axis. However, because the graph of cosine is symmetric, this has no effect at all. So the graph and the function match.
If wrong: Incorrect. This graph has the correct amplitude and period, but it is the graph of a sine function. In particular, this graph passes through , while the graph of does not.
If wrong: Incorrect. This graph has the correct amplitude and period. However, it appears that you confused the effect of a negative sign on the inside with a negative sign on the outside. The effect of the negative sign is to reflect the graph around the y-axis. However, because the graph of cosine is symmetric, this has no effect at all.

Q: Which of the following functions is represented by the graph below?
A: (a choice with no text)
If wrong: Incorrect. The graph does have the shape of a sine function, and you found the correct amplitude, but the period is incorrect. Perhaps you recognized that the period of the graph is one fourth the period of , and thought that would make . But to find the value of b, you must set the period equal to .
If wrong: Incorrect. The graph does have the shape of a sine function, and you found the correct amplitude, but the period is incorrect. Perhaps you observed that the graph has 6 cycles and used this to choose your answer. But note that these 6 cycles are shown on the interval . There are only 4 cycles on the interval .
If wrong: Incorrect. You seem to have confused the roles of a and b. The amplitude of is 4, while the amplitude of the graph is 3. The period of is , while the period of the function in the graph is .
If right: Correct. This graph has the hill and valley shape of a sine function, so . The amplitude is 3. It has the same orientation as the graph of and therefore . Now there are 6 cycles shown on the interval , or 4 cycles on the interval . This tells you that . So this could be the graph of .