Sohcahtoa

Sohca...what? Just an easy way to remember how Sine, Cosine and Tangent work:

Soh...
Sine = Opposite / Hypotenuse
...cah...
Cosine = Adjacent / Hypotenuse
...toa
Tangent = Opposite / Adjacent

Right Triangle

OK, let's see what this is all about.

Firstly, the names Opposite, Adjacent and Hypotenuse come from the right triangle:

triangle showing Opposite, Adjacent and Hypotenuse

  • "Opposite" is opposite to the angle θ
  • "Adjacent" is adjacent (next to) to the angle θ
  • "Hypotenuse" is the long one

Opposite, Adjacent and Hypotenuse

Adjacent is always next to the angle
(and opposite is opposite the angle)

Sine, Cosine and Tangent

And Sine, Cosine and Tangent are the three main functions in trigonometry.

They are often shortened to sin, cos and tan.

The calculation is simply one side of a right angled triangle divided by another side ... we just have to know which sides, and that is where "sohcahtoa" helps.

For a triangle with an angle θ , the functions are calculated this way:

Sine Function:
soh...
sin(θ) = opposite / hypotenuse
Cosine Function:
...cah...
cos(θ) = adjacent / hypotenuse
Tangent Function:
...toa
tan(θ) = opposite / adjacent

 

Example: what are the sine, cosine and tangent of 30° ?

The classic 30° triangle has a hypotenuse (the long side) of length 2, an opposite side of length 1 and an adjacent side of √3, like this:

30° triangle

Now we know the lengths, we can calculate the functions:

Sine
soh...
sin(30°) = 1 / 2 = 0.5
Cosine
...cah...
cos(30°) = 1.732 / 2 = 0.866
Tangent
...toa
tan(30°) = 1 / 1.732 = 0.577

(get your calculator out and check them!)

How to Remember

Well, "sohcahtoa" may be easy for you to remember ... but here’s another way to help it:

Sailors Often Have Curly Auburn Hair Till Old Age.

Or perhaps you prefer one of these:

  • Some Old Horses Can Always Hear Their Owners Approach.
  • Some Old Hen Caught Another Hen Taking One Away.