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.

First, 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

examples of Opposite, Adjacent and Hypotenuse

Adjacent is always next to the angle

And Opposite is opposite the angle

Sine, Cosine and Tangent

The main functions in trigonometry are sine, cosine and tangent.

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: soh... sin(θ) = opposite / hypotenuse

Cosine:...cah...cos(θ) = adjacent / hypotenuse

Tangent:...toatan(θ) = opposite / adjacent

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

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

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

Sine: soh... sin(30°) = 12 = 0.5

Cosine:...cah...cos(30°) = 1.732...2 = 0.866...

Tangent:...toatan(30°) = 11.732... = 0.577...

(get your calculator out and check them!)

How to Remember

I find "sohcahtoa" easy to remember ... but you might like these:

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.

Have a practice here:

../geometry/images/triangle-q.js?mode=cst

And try these questions:

728, 1496, 729, 1497, 2364, 2365, 2366, 2367, 2368, 2369