Trigonometry Study Hall

Trig Basics

Despite its reputation as being "high math", trigonometry can be very simple. At least its use on the SAT or ACT is simple—and that's all we're worried about here.

To use trig, you need two things:

1. a triangle with a right angle (AKA, a right triangle)
2. one of the other parts labeled (either an angle or a leg-length)

If you have these two things, you can figure out the rest of the parts fairly easily.

Look at the graphic above

• There are 3 angles (of course): $\angle A$ , $\angle B$ , and $\angle C$ . Note that $\angle C$ is the right angle.
• There are 3 legs or sides (of course): $a$ , $b$ , $c$
• The leg-labels are "Opposite", "Adjacent", and "Hypotenuse"
  • The "Hypotenuse" will always be the same, it's the hypotenuse! "Opposite" and "Adjacent" will change depending on which angle we're talking about.
  • The labels shown are in relation to $\angle A$. We know this because starting with $\angle A$, we can see that the "Adjacent" leg is indeed adjacent (next to it) to the angle. And the "Opposite" leg is directly opposite from the angle.

Basic Trig Functions

The chart above should be read such as $sinx={opp \over adj }$ meaning the $sin$ of $\angle x$ is equal to the opposite side divided by the adjacent side.

An Example

Suppose we knew that side b was 32 feet and we're asked to find out the length of side C (the hypotenuse).

1. First, take a look at what we know. We know…
   1. $\angle C = 90^\circ$
   2. Side b is 32 feet.
2. We have two parts, that's all we need.
3. Next, decide if we want to use $sin$, $cos$, or $tan$.
4. Focus on $\angle C$ because that's the one we know. We know the adjacent side is 32 and want to know the hypotenuse.
   1. So, we're dealing with Adjacent and Hypotenuse. The Trig Functions chart tells us Adjacent and Hypotenuse go with Cosine.
5. Our formula would then look like the one below…

$cos90={32 \over hyp}$

From here, it's just a matter of punching in some numbers on a calculator to get the cosine of 90, then doing some simple algebra to solve for the hypotenuse. The steps would be…

Punch in 32 and hit "cosine" on your scientific calculator. It'll show you 0.848. The formula would then read…

$0.848={32 \over hyp}$

To solve for $hyp$, cross multiply like this: $0.848 \times hyp = 32 \times 1$ and you'll get…

$0.848 hyp = 32$

To solve, simply isolate the variable ($hyp$) by dividing both sides by 0.848

$32 \div 0.848 = 37.735$ and that's your answer. The hypotenuse is $37.735$ feet long.

A note

If we needed to go farther, we could. We could now use the Pythagorean Theorem ($a^+b^2=c^2$) to find the other side's length. Combing Trig and Pythag, we can find all of the angles and leg-lengths of a right triangle from only two initial measurements.

Mnemonic Devices for Trig

Below are some tricks to help remember the basic trig functions.

1. SohCahToa — the ancient Indian princess of geometry.
   • This puts the parts in order (S=sin, C=cosine, T=tangent, o=opposite, h=hypotenuse, a=adjacent)
2. Sally can tell Oscar has a hat on always.
   • This sentence puts the parts in a different order.