Most junior class math students in America know what SOH CAH TOA stands for:
The Sine (of an angle) = the Opposite divided by the Hypotenuse.
The Cosine (of an angle) = the Adjacent divided by the Hypotenuse.
The Tangent (of an angle) = the Opposite divided by the Adjacent.
They also know these trigonometry rules apply to right triangles only and not non-right triangles. (See the diagram above)
If they are solid with these rules or just need to brush up on their skills, good things await for their ACT math scores...
The ACT Math Section sometimes contains as many as three SOH CAH TOA questions. That's potentially three nearly free points for a test taker. And remember that for an ACT Math Section three points can be huge. A few more correct answers in the ACT Math Section can raise a scale score a whole point and sometimes two!
TEST-TIP: MEMORIZE the SOH CAH TOA rules and practice them for your ACT
(But...always remember to guess quickly on any question--even a SOH CAH TOA question--that stumps you, then move on to an easier one!)
To all of you ACT test takers out there, let's review SOH CAH TOA problems with an eye for how the ACT likes to present them.
As many of you know, the math section gets harder as you go from question 1 to question 60. And SOH CAH TOA problems can appear near the beginning, in the middle, or near the end of an exam. In other words they can be easy, medium in difficulty, or tough. But if you know the trig rules well and have had a little practice, even tougher end-of-the-test versions can be manageable. Also, most of the ACT's SOH CAH TOA problems come with a figure drawn for you. All you need to do is the final solving steps using your trig rules and algebra skills.
EASY SOH CAH TOA:
An easy SOH CAH TOA (SCT) problem could look something like this problem pictured below.
But before we get to it, in general, follow these basic steps for SCT probs:
1. find your known angle, your known side and another known or unknown side
2. set up an equation using either SOH, CAH, or TOA
3. solve your equation
For this problem above you do not have use your calculator. As well, which trig rule to use is already revealed by the question itself. All you need to do is find your known and unknown sides and which angle to use.
The angle part is easy because the problem itself asks for the sine of angle A.
Therefore, we need to use the trig rule for sines which is "SOH". Now, simply look at where angle A is in the triangle. Then, from that corner of the triangle ask yourself which side is opposite of angle A...It is 12 km. Then ask what is the length of the hypotenuse side (always the longest side)...It is 13 km. Follow the SOH rule now. The sine of angle A is the opposite (12 km) over the hypotenuse (13 km) to make the fraction (or ratio) of 12/13. The correct answer is C! Not too bad right? Go step-by-step for an easy point.
MEDIUM SOH CAH TOA:
A semi-tough SCT question could appear around the middle of the exam and look like the next problem below. How would you know this might be a SCT problem? Two big clues are that there is a right triangle involved, and the answers themselves contain trig words like tan, cos and sin. To solve this problem, you will not only have to correctly find two sides (known and unknown) to satisfy one of the three trig rules (as in the problem above), but you will have to study the figure to decide which rule to use in the first place. You will also have to do a little algebra! But if you go step by step, it's not too bad.
First read the problem carefully and try to figure out what it is asking you to find. Hopefully, it's clear enough to see that you need to find the distance from "the boat to the dock". This is your unknown side. Label it x on the diagram.
x = distance from boat to dock.
But what angle should you use? For the angle, use the only other corner besides the right angle where you know the degree value. It's the lighthouse corner angle which measures 52 degrees. This is your known angle.
Now, which trig rule should you use: SOH, CAH, or TOA? To find out, ask which side is your unknown side labelled x. Is it opposite of the 52 degree angle or adjacent to it? Or is it the hypotenuse (longest side) of the triangle. ANSWER: It's the opposite.
Now ask which side is the side that you do have a measurement for already. It's the side for the "dock to the lighthouse" labelled with a distance of 30 miles. This is your known side and it is adjacent (or next to) the lighthouse corner.
At this point you are ready to go! You have an unknown opposite side--labelled "x"--and a known adjacent side of 30 miles and a known angle of 52 degrees.
Therefore you would use the TOA rule to make a solvable equation. Write the algebra equation in your work space, remembering that the tangent of a known angle = the opposite divided by the adjacent.
tan(52) = x/30
Now solve it by using your algebra skills to get "x" alone.
(30) tan(52) = x/30 (30)....multiply both sides 30
30tan(52) = x....................the 30's cancel on the right side
F is the right answer! Note: you don't need your calculator for this question, just regular trig and algebra skills.
TOUGH (CHALLENGING) SOH CAH TOA:
For an example of a really tough SCT problem found near the end of the test, see the problem below. If you want to challenge yourself see if you can do it alone, then check your answer below!
Before tackling this problem, recall that this tough problem is not worth stressing about when it comes to your overall strategy. For the ACT Math Section, always do easier problems first! However, if you are a strong SCT solver and feel like you know what is going on...and you have ample time near the end of the test, take a crack at something like this!
To solve this you need to know SOH CAH TOA rules and how they work for inverse trig relationships. The symbol for inverse trig stuff is that little exponent of -1 that hovers above the tan expression in the problem.
Step one is recognize that this is a SCT based problem because you see a right triangle and trig words like cos and tan in the problem.
Step two is quickly dig back into your memory of trig lessons from your higher algebra classes and recognize that this problem is all about using the figure to find an angle and then using SCT to find a ratio.
If you don't have that memory SKIP THIS PROBLEM right away and make a quick guess! It's only worth one point, like all ACT problems.
If you do, let's move forward.
Step three is use the figure to find the unknown angle. Work inside-out on the given expression which is cos [tan^-1(a/b)].
The "inside" of the expression is tan^-1(a/b), "the inverse tangent of a over b".
What angle does this inverse tangent expression point to? From the figure you can see that the bottom right corner angle would be the angle that the inverse tangent of a/b refers to because from this corner the tangent of that angle would be "a over b" or the "opposite over the adjacent".
[If this is terribly confusing, remember you could always guess quickly on this question!...if you still want to understand this ask a math tutor or teacher or math-happy friend!]
Step four...Moving on, now do the "outside". In other words find the cosine of the angle you just located. The cosine of that bottom right angle is the ratio you create from the trig rule for cosine: CAH.
Finally, the adjacent side to this corner angle is "b" and, using CAH, it goes in numerator above the hypotenuse, which is the square root of a^2 + b^2.
So the answer is D!
The ACT loves SOH CAH TOA problems!
So it will be worth your time to master the rules and to practice a good number of actual ACT example problems of this kind. However, if the thought of doing trigonometry of this kind really stresses you out, find some tutoring help and craft a strategy that works for you. Depending on your math background and time to prepare, it might mean taking quick guesses on these kinds of problems, really mastering them, or somewhere in between. Hope these math tips help raise your next math section score!