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Kristin Fracchia

TuesdACT: Trigonometry Basics — Sine, Cosine, and Tangent

Many, many, MANY of the students I have tutored for the ACT are intimidated by the fact that there is trigonometry on the test. They either haven’t studied it yet, have barely started studying it, or are feeling lost in their current trig classes. But the good news is that the trig you need to know is, for the most part, very basic, and often I’ve just taught the few things you do need to know to students as part of their ACT tutoring because they are so relatively simple, and you can pick up points by just knowing a few things. So let’s start with the basics and then we we will get a little more advanced in future posts.

First of all: What is trigonometry?

Trigonometry is the field of math that deals with triangles–specifically, the relationships between the three sides and the three angles that make up every triangle.

And typically the first thing you study in a trig class are right triangles:

Screen Shot 2015-11-05 at 5.06.08 PM

So here’s a right triangle. Let’s say that we are looking at the angle the arrow is pointing to. The side next to it is the adjacent side, the side opposite it is the opposite side, and the hypotenuse is, of course, the hypotenuse.

It is important that you think about the sides this way because the next thing you typically learn in a trig class is a mnemonic called SOHCAHTOA, and these As and Os and Hs stand for adjacent, opposite, and hypotenuse.

But what do the S, C, and T stand for?

The next things you need to memorize about trig are these three terms and their abbreviations:

    Sine (sin)
    Cosine (cos)
    Tangent (tan)


These three terms are used to designate the ratio of a pair of sides in a triangle.

So here is where SOHCAHTOA comes in. This helps you remember which ratio is which:

    sin → opposite/hypotenuse
    cos → adjacent/hypotenuse
    tan → opposite/adjacent


I suggest anytime you see a right triangle with trig terms on the ACT that you write down SOHCAHTOA next to the problem because it’s very easy to accidentally use the wrong ratio.

Example 1:

What is the sin of A?

Screen Shot 2015-11-05 at 5.01.02 PM

Knowing SOHCAHTOA, you would be able to answer that it is opposite/hypotenuse or 3/5 . Easy as that!

Example 2:

What is the length of XZ?

Screen Shot 2015-11-05 at 5.01.25 PM

Knowing SOHCAHTOA means that if we are given a right triangle with one known length and one known acute angle (meaning not the right angle) we can always find the other two lengths.
So in this case we can use sine to find the length of the hypotenuse.

sin(10) = 3 / XZ
XZ = 3 / sin (10)

We can divide sin of 10 degrees by 3 in our calculator to get the answer: approximately 17.28.

Example 3

Here’s one that’s just a teensy bit harder, but we are just going to apply the same principles.

The tree below casts a shadow that is 24 feet long, and the angle of elevation from the tip of the shadow to the top of the tree has a cosine of 4/5. What is the height of the tree?

Screen Shot 2015-11-06 at 5.04.59 PM

The problem tells us that the cosine of the angle of elevation is 4/5. Remember SOHCAHTOA, so we are concerned with the adjacent side over the hypotenuse. The fact that the cosine is 4/5 means the ratio of the adjacent side to the hypotenuse is 4/5. So we can set up a proportion:

4/5 = 24/x

Cross-multiplying to solve for x gives us x = 30.

But remember that this is the hypotenuse and we need to find the length of the vertical side to find the height of the tree. We can use the Pythagorean Theorem to find the length of the vertical side.

24^2 + b^2 = 30^2


So the height of the tree is 18 ft.

If you recognized that we had a 3-4-5 triangle in the beginning, you could actually take a shortcut and just use tangent of the angle of elevation to figure out the height.

Knowing SOHCAHTOA just about guarantees you will be able to nab at least one trig question on the ACT. And we’ll share more trig basics in future posts!

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About Kristin Fracchia

Kristin makes sure Magoosh's sites are full of awesome, free resources that can be found by students prepping for standardized tests. With a PhD from UC Irvine and degrees in Education and English, she’s been working in education since 2004 and has helped students prepare for standardized tests, as well as college and graduate school admissions, since 2007. She enjoys the agony and bliss of train running, backpacking, hot yoga, and esoteric knowledge.

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