Trigonometric Ratios

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

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

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

Adjacent is always next to the angle (andopposite 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 ... you 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 1and an adjacent side of √3, like this:

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!)

The other ratios are simply inverses of the above three ratios:

 co-secθ = 1/sinθ

secθ = 1/cosθ

cotθ = 1/tanθ

How to Remember

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

Sailors Often Have Curly Auburn Hair Till Old Age.