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The Tricks to Trigonometry

Updated: Jan 7, 2023



(Cover Artist: Sophie Cheng )

“Trigonometry” sounds like a scary word. I know that when I was little, I would see the word somewhere or hear it spoken and would classify it in the “scary math” category along with calculus and other long, “mathy” words. I never thought of trigonometry as something I would be seeing in my near future. But here we are, as I write this article about how trig is simpler than you might think.

Trigonometry starts with the idea of right triangles. It is used to find unknown sides, or angles of a shape. That's about as difficult as this article goes, in order to create a concrete solid foundation and introduction to this information.

Let’s first have a refresher on what a right triangle is. A right triangle is a triangle with one right angle, meaning one 90 degree angle. In a diagram, the angle with a little box on it is the right angle showing the observer of the diagram that the two lines of the angle are perpendicular.


The hypotenuse is the side length of the triangle that sits directly across from the right angle. So for example in this diagram, side length C is the hypotenuse.


SINE

Finally, onto trig. Let’s start with sine. Sine means Angle=Opposite/Hypotenuse

If we are given an angle measure, for example 50, like in this diagram. We are also given one of the side measures for sine, they hypotenuse and the side measure we are trying to find, represented by x, is the other side measure needed for sine. This means we can calculate x by using sine. So let’s plug in each side measure and angle.

sin(50) = x/9

To calculate, we can now multiply 9 by the sine of 50, and find our answer for x. This can be calculated using a calculator.

x = 9sin(50)

The result is going to be x.


We can use the same process for each different function.


COSINE

The equation for cosine is

cosine of the angle = adjacent side/hypotenuse

(*Note: the " / " you see represents division, and does not represent "or".)



Following the function, we can plug in the angle= the adjacent side length(the side length next to the angle) over the hypotenuse.

cos (35) = a/20

Now, we can solve this by multiplying both sides by 20, and find the measure for the missing side of a.

a = 20cos(35)


TANGENT

Lastly, let's talk about tangent. Also known as tan, tangent is the special function because it does utilize the hypotenuse.

The equation for tan is

tangent of an angle = opposite side/adjacent side

In this triangle, we plug in 20 as our angle and put 10/x to plug in for the opposite of adjacent.

tan(20) = 10/x

To solve this we can multiply x times tan of 20 and then divide the tan of 20 from x as well as tan to put x by itself.

x = 10tan(20)

This will leave us with the result of side x.


TIPS AND TRICKS

An easy way to remember what the equation is for each function you can remember the acronym Soh Cah Toa.

The capital letters stand for the function and the lower case stands for either opposite, adjacent, or hypotenuse.

"Soh" is Sine = opposite side/hypotenuse

"Cah" is Cosine = adjacent side/hypotenuse

"Toa" is Tangent = opposite side/adjacent side


This chart should be useful !

Keep in mind that for now, you can't use the right angle as your base angle; you must use a different, acute angle.

Finally, attached below are some example triangles to practice what we've learned. Try finding the sine, cosine, and tangent of each triangle!




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