Tip #35 to Get a Top ACT Math Score (page 2)
Half of the trig questions on the ACT are just SohCahToa, which you've now mastered. Many kids think that the rest of the trig questions are way beyond them, that you can only get them if you've had a full-year trig course. If you've had a trig course, that's great, and these questions will be especially easy for you. If not, the exciting news is that almost every trig question can be done with "Use the Answers" or "Make It Real"! You'll see examples of this when we review the Pretest question and in the drills.
When the ACT wants you to use a specific trig concept, such as the law of sines or law of cosines, they will explain it in the question, and you just have to follow the directions. These are great; they give you the equation, and you just follow the directions. You'll see this also in the drills.
Lastly, occasionally the ACT has a question about the reciprocals of sin, cos, and tan, which are
Let's look at this question:
Solution: θ is just the symbol that people use for an angle when it's unknown. (The symbol is called theta.) Therefore, "what is the value of θ" means "what is the angle measure?" You probably could have guessed that anyway, since the question tells you that θ is between 0 and 360. Another example of "Don't get intimidated!" Just go with it, make an assumption on the ACT, and you'll probably be correct. They set it up that way. This is a great example of a Beyond SohCahToa question where you really don't need any trig. You just need to "Use the Answers"! Using your calculator, try each choice for θ and see which one gives you sin θ = –1. Choice D is correct, because sin 270 = –1. If using the "sin" button is new for you, practice it right now. It's easy. Just hit "sin" and type "270" and hit Enter and you'll get an answer of –1.
Correct answer: D
- What are the values of θ, between 0° and 360°, when tan θ = 1 ?
- 45°, 135°, 225°, and 315°
- 45° and 135° only
- 45° and 225° only
- 45° and 315° only
- 135° and 315° only
- Which of the following expressions gives the perimeter of the triangle shown below, with measurements as marked?
- Which of the following equations reflects the graph shown below?
- y = sin x
- y = 5 sin (x - 2)
- y = 2 cos x - 2
- y = cos (x - 2) - 2
- y = 2 tan x - 5
- If b, c, and d represent positive real numbers, what is the minimum value of the function f(x) = sin b(x - c) - d ?
- d + c
- -1 - d
(Note: The law of sines states that the ratios between the length of the side opposite any angle and the sine of that angle are equal for all interior angles in the same triangle.)
- C Awesome "Use the Answers" question! Try each answer choice, and use the process of elimination until you find the 1 choice that has values that all work. Choice C is correct since both 45 and 225 yield 1 when plugged in for #&952; in the expression tan #&952;. If you've studied trig, you could also do this question the "math class way." Since tan means "opposite over adjacent," tan #&952; = 1 when opposite = adjacent, and therefore sin = cos. So what are the values for #&952; where sin = cos? Sin could equal cos when #&952; is between 0 and 90 or between 180 and 270, because in these regions sin and cos are either both positive or both negative. Use the process of elimination, and only choice C has answers in these two regions.
- F Follow the directions given for the law of sines. The ratio of a side and the sin of its opposite angle is equal for all sides. So Perimeter equals the sum of the sides of the triangle, and we already know two of the sides. We can use that ratio to solve for x, the third side. So since we can cross-multiply to get (x)(sin 74) = (32)(sin 52), and divide both sides by sin 74 to get Therefore, the perimeter equals
- D Notice that all of these are "hards." You would only see this Beyond SohCahToa stuff as hards. You can "Use the Answers" here. Graph each answer choice on your calculator, and find the one that matches the graph shown in the question. You'll notice, when you graph these, that they don't make the curved line like in the picture; that's your tipoff that you need to change your calculator to radians mode. That's the only curveball for this particular type of question. You need to change the mode on your calculator to radians, and you have to remember to switch it back when you're done. Just ask your math teacher to show you this if you've never done it before. Radians are just another way (besides degrees) to measure an angle. Once you know how to do this on your calculator, choice D matches perfectly.
- K This question is crazy theoretical, so you can "Make It Real." Just choose positive numbers for b, c, and d and graph the equation on your calculator. Then find the minimum value (the lowest point) of the graph. You'll notice when you graph these that they don't make the usual trig curved graph like the diagram in question 3; that's your tipoff that you need to change your calculator to radians mode. That's the only curveball for this particular type of question. You need to change the mode on your calculator to radians, and you have to remember to switch it back when you're done. Just ask your math teacher to show you this if you've never done it before. Radians are just another way (besides degrees) to measure an angle, and once you know how to do this on your calculator, it's easy.
If you've studied trig, you could also do this question the "math class way." The graph shown has been shifted from its usual position. When a graph has been shifted, we can use the equation y = a cos b(x – c) – d, where a tells the altitude of the curve (how high up and down it reaches), b/(2π) tells the period (the length of one repeat), c tells how far left or right the graph was moved from the origin, and d tells how far up or down the graph was moved from the origin. The graph in the question has been shifted down and to the right from the ordinary cos graph. So we want a cos graph with a right and down shift, represented by the c and d. So answer choice D is correct. It is the only one with a number for c and d in the equation y = a cos b(x – c) – d.
To "Make It Real," let's say b = 2, c = 3, and d = 4; then the graph shows a minimum value of –5. Using the numbers we choose for b, c, and d in the answer choices, we need the choice that also yields –5. Choice K works, since –1 – 4 = –5.
If you've studied trig, you could also do this question the "math class way." In the equation y = a cos b(x – c) – d, the c tells how far left or right the graph was moved from the origin, and d tells how far up or down the graph was moved from the origin. So this graph will always reach a minimum of –1 – d, since it's normal minimum is –1 and it has been shifted down d units, giving a new minimum of –1 – d.
Go to: Tip #36
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