Step by step
The question
Question: find the angle θ of a right triangle with opposite side 7 and adjacent side 11.
1 Name the sides
θ is a letter for the angle we want.
Say it "theta".
We look from there.
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
For one angle, every right triangle has the same shape.
Only the size changes.
So each pair of sides always gives the same number.
The calculator knows these numbers.
They are called sin, cos and tan.
SOH CAH TOA helps us remember.
Sin is O over H.
Cos is A over H.
Tan is O over A.
We have the opposite and the adjacent: O and A.
TOA has them.
So we use tan.
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: O = 7 and A = 11.
7 divided by 11 is about 0.6364.
4 Turn it into the angle
The tan⁻¹ button on the calculator turns a ratio back into the angle.
The small −1 does not mean one over.
It means: go back.
It asks: which angle has a tan of 0.6364?
So θ = tan⁻¹(0.6364).
The calculator gives about 32.471.
To the nearest tenth of a degree, θ ≈ 32.5°.
5 Check
Let's check: tan of 32.5° on the calculator.
0.6364 again, nearly: the small gap is from rounding the angle.
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.