Whether you're brushing up on your trigonometry for the SAT or ACT or learning how to calculate angles for the first time, SOHCAHTOA is a useful mnemonic device to know. In this article, we'll explain what SOHCAHTOA is, how to use SOHCAHTOA, and give SOH CAH TOA examples.
What Does SOHCAHTOA Mean?
SOHCAHTOA is an easy way to remember how to calculate different functions in trigonometry.
The three different functions you can calculate using SOHCAHTOA are sine, cosine, and tangent. Sometimes, these functions are shortened to sin, cos, and tan.
You can use SOHCAHTOA to remember how to calculate each of these trigonometric functions in a right triangle.
SOHCAHTOA tells you that:
SOH
$\bo{S}\ine = {\bo{O}\pposite} / {\bo {H}\ypotenuse}$
CAH
$\bo{C}\osine = {\bo{A}\djacent} / {\bo{H}\ypotenuse}$
TOA
$\bo{T}\angent = {\bo{O}\pposite} / {\bo{A}\djacent}$
Understanding What SOHCAHTOA Means
Let's expand on what we covered in the section before with an example. Remember, in order to use SOHCAHTOA, you need to know what each letter in the acronym refers to:
- S: Sine
- O: The side opposite from the unknown angle
- H: The side opposite to the right angle (the longest in the triangle)
- C: Cosine
- A: The side next to the unknown angle
- T: Tangent
Here's an example:
In the above example O is the angle you're trying to solve for.
The side opposite from the unknown angle is 3.
The side adjacent (touching) the unknown angle is 4.
The hypotenuse (across from the right angle) is 5.
You need to know which side is which to solve problems using SOHCAHTOA. You don't need to know their lengths ahead of time: that's what you'll be solving for!
SOHCAHTOA Calculator In Action
Now that we know what we need to solve a SOHCAHTOA problem, let's put that into action using the same example.
We can solve for SINE, COSINE, and TANGENT using the information on this shape. Follow these steps to do so:
#1: Identify the Sides
We've done this already!
The side opposite from the unknown angle is 3.
The side adjacent (touching) the unknown angle is 4.
The hypotenuse (across from the right angle) is 5.
#2: Write SOHCAHTOA on Your Paper
This helpful step will make sure you don't forget one of the formulas.
#3: Use SOHCAHTOA to Solve
Let's solve SINE first, using the side we've already identified.
$\SINE = {\opposite}/{\hypotenuse}$
$\SINE = 3/5$
Now, let's solve COSINE.
$\COSINE = {\adjacent}/{\hypotenuse}$
$\COSINE = 4/5$
Finally, let's solve TANGENT.
$\TANGENT = {\opposite}/{\adjacent}$
$\TANGENT = 3/4$
That was easy!
Using SOHCAHTOA to Find Side Lengths
You can also use SOHCAHTOA to find side lengths. Let's look at how.
We know that the angle is 60 degrees. We're given the hypotenuse (the side across from the right angle). We are being asked to find X, which is the side adjacent to the angle. That means we need to use cosine.
So, we have:
$\cos60 = \X/10$
To solve for X, we multiply both sides by 10:
$(\cos60) * 10 = (\X/10 * 10)$
$9.5=\X$
SOHCAHTOA Calculator: Examples
Let's put our SOHCAHTOA knowledge into practice.
SOHCAHTOA Calculator: Example 1
SINE =
COSINE =
TANGENT =
Answers:
$\SINE = 4/3$
$\COSINE=3/5$
${\TANGENT}=4/5$
SOHCAHTOA Calculator: Example 2
SINE =
COSINE =
TANGENT =
Answers:
$\SINE = 3/5$
$\COSINE =4/5$
${\TANGENT} = 3/4$
SOHCAHTOA Calculator: Example 3
What is the value of X?
Answer:
$\sin(50) = \X/7$
$\sin(50) * 7 = (\X/7) * 7$
$\X = 3.9$
Tips for Using SOHCAHTOA
Still not quite sure how to use SOHCAHTOA? Here are some tips.
#1: Remember the Correct Order
SOHCAHTOA is easy to use, but you need to make sure you're using it correctly. If you get any of the letters mixed up, you will be unable to use the formula correctly.
Here are some fun tongue twisters you can use to remember SOHCAHTOA:
- Sailors Often Have Curly Auburn Hair Till Old Age.
- Some Old Horses Can Always Hear Their Owners Approach.
- Some Old Hen Caught Another Hen Taking One Away.
#2: Write Down the Formula
Whether you're working on the SAT or just practicing trigonometry on your homework, writing down SOHCAHTOA will help you remember how to use it. That way, you won't get mixed up!
#3: Identify Your Sides First
Remembering which side is opposite and which side is adjacent can be difficult. You should identify each side and write that on your paper first so you don't make any careless errors while you're working.
Final Thoughts: How to Use SOHCAHTOA
What's the SOHCAHTOA meaning? SOHCAHTOA is a useful mnemonic device to remember how to calculate the lengths of sides and angles in right triangles. Brushing up on the SOH CAH TOA formula and trying out SOH CAH TOA problems can help you get ready for your SAT or ACT.
What's Next?
Make sure you're prepared for SAT/ACT Math by reading our Ultimate Guide to SAT Math Prep and our Ultimate Guide to ACT Math Prep!
If you're feeling pretty confident, why not test your skills against the 13 hardest SAT Math questions ever? If you can conquer those, you'll likely do great on the SAT, too! (Here are the hardest math questions on the ACT, too.)
If you need a little extra practice, why not check out some math prep books? Here's a list of our favorites.
