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# Easy Test-Taking Tips: Know Your SOH CAH TOA for the Math Section of the ACT! (Plus a challenge ACT math problem!)

July 22, 2019

Most junior classÂ math students in America know what SOH CAH TOA stands for:

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The Sine (of an angle) = the Opposite divided by the Hypotenuse.

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The CosineÂ (of an angle) = the AdjacentÂ divided by the Hypotenuse.

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The Tangent (of an angle) = the Opposite divided by theÂ Adjacent.Â

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They also know these trigonometry rules apply to right triangles only and not non-right triangles. (See the diagram above)

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If they are solid with these rules or just need to brush up on their skills, good things await forÂ their ACT math scores...

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The ACT Math Section sometimes contains as many as three SOH CAH TOA questions. That's potentially three nearly free points forÂ a test taker. And remember thatÂ for an ACT Math SectionÂ threeÂ points can be huge.Â A few more correct answers in the ACT Math Section can raise a scale score a whole point and sometimes two!

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TEST-TIP:Â MEMORIZE the SOH CAH TOA rules and practiceÂ themÂ for your ACT

(But...always remember to guess quickly on any question--even a SOH CAH TOA question--thatÂ stumps you, thenÂ move on to an easier one!)

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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.

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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 anÂ 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'sÂ SOH CAH TOA problems come with a figure drawn for you. All you need to do is the finalÂ solving steps using yourÂ trig rules and algebra skills.

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EASY SOH CAH TOA:

An easy SOH CAH TOA (SCT) problem could lookÂ something like this problem pictured below.

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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,Â CAH, or TOA

3. solve your equation

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For this problem above you do not have use your calculator. As well,Â which trig rule to use is already revealed by the question itself. All you need to do is find your known and unknownÂ sides and which angle to use.

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The angle part is easy becauseÂ the problem itselfÂ asks forÂ the sine ofÂ angleÂ A.Â

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Therefore,Â we need to use the trig rule for sines which isÂ "SOH". Now, simply look at where angle A is in the triangle. Then, from that corner of the triangle ask yourself whichÂ side is opposite of angle A...ItÂ is 12 km. Then ask what is the length of theÂ hypotenuse side (always the longest side)...It is 13 km. Follow the SOHÂ rule now. The sine of angle A is the opposite (12 km) over the hypotenuse (13 km)Â to make the fraction (or ratio) ofÂ 12/13. The correct answer is C! Not too bad right? Go step-by-step for an easy point.

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MEDIUMÂ SOH CAH TOA:

A semi-tough SCT questionÂ could appear around the middle of the exam and look like the next problem below. How would you know thisÂ might be a SCT problem?Â Two big clues are that there is a right triangle involved, and the answers themselves contain trig words likeÂ tan, cos and sin. To solve this problem,Â you will not only have to correctly find two sides (known and unknown) to satisfyÂ 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 theÂ first place. You will also have to do a little algebra!Â But if you go step by step,Â it's not too bad.

First read the problem carefully and try to figure outÂ what it is asking you to find. Hopefully, it's clear enough to seeÂ that you need to find the distance from "the boat to the dock". This is your unknown side.Â Label it x on the diagram.

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x = distance from boat to dock.

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But what angle should you use? For the angle, useÂ the only other corner besides the right angle where you know the degree value.Â It's the lighthouse corner angle which measuresÂ 52 degrees. This is your known angle.

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Now, which trig rule should you use: SOH, CAH, or TOA? To find out, ask which side is your unknown side labelled x. Is itÂ opposite of the 52 degree angle orÂ adjacent to it? Or is it the hypotenuse (longest side) of the triangle. ANSWER:Â It's the opposite.

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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 itÂ is adjacent (or next to) the lighthouse corner.

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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.

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Therefore you would use the TOAÂ rule to make a solvableÂ equation.Â Write theÂ algebraÂ equation in your workÂ space, rememberingÂ thatÂ the tangentÂ of a known angle = the opposite divided byÂ the adjacent.
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tan(52) = x/30

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Now solve it by using your algebra skills to get "x" alone.

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(30) tan(52) = x/30 (30)....multiply both sides 30Â

30tan(52) = x....................the 30's cancelÂ on the right side

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F is the right answer! Note: you don'tÂ need your calculator for this question, just regularÂ trig and algebra skills.

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TOUGH (CHALLENGING)Â SOH CAH TOA:

For an example of aÂ really tough SCT problem found nearÂ the end of the test, see the problem below. If you want to challengeÂ yourself see if you can do it alone, then check your answer below!

Before tackling this problem, recall that this toughÂ problem is not worth stressing about when it comes to your overall strategy. For the ACT Math Section, always do easier problems first!Â 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!

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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.

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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.

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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.

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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.

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If you do, let's move forward.

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Step three is use the figure to find the unknown angle. Work inside-out on the given expression which isÂ cos [tan^-1(a/b)].

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The "inside" of the expression isÂ tan^-1(a/b), "the inverse tangent of a over b".Â

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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".

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[If this is terribly confusing, remember you could always guess quickly onÂ this question!...if you still want to understand this ask a math tutor or teacher or math-happy friend!]

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Step four...Moving on, now do the "outside".Â In other words findÂ the cosine of the angle you just located. The cosine of that bottom rightÂ angle is the ratio you create from the trig rule for cosine: CAH.Â

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Finally, the adjacent side to this corner angleÂ is "b" and, using CAH, it goes in numerator aboveÂ the hypotenuse, whichÂ is the square root of a^2 + b^2.

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So the answer is D!Â

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SUMMARY:

The ACT loves SOH CAH TOA problems!

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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 kindÂ 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,Â really mastering them, or somewhere in between. Hope these mathÂ tips help raise your next math section score!

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