Step by step
The question
Question: find the angle θ of a right triangle with adjacent side 6 and hypotenuse 12.
1 Name the sides
We look from the angle θ.
Every side gets a name from there.
The hypotenuse H is the longest side.
It is across from the right angle.
The opposite side O is across from our angle.
The adjacent side A is next to our angle.
It is not the hypotenuse.
2 Choose the ratio
SOH CAH TOA helps us remember.
Sin is O over H.
Cos is A over H.
Tan is O over A.
We have the adjacent and the hypotenuse: A and H.
CAH has them.
So we use cos.
Careful!
From the other angle, the opposite and adjacent sides would swap.
Always name them from the angle we use.
3 Write the equation
Now put the numbers in: A = 6 and H = 12.
6 divided by 12 is 0.5.
4 Turn it into the angle
The cos⁻¹ button on the calculator turns a ratio back into the angle.
So θ = cos⁻¹(0.5).
The calculator gives exactly 60.
So θ = 60°.
5 Check
Let's check: cos of 60° on the calculator.
That is 0.5 again.
So θ is right!
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: turn it into the angle.
Last: check.