Step by step
The question
Question: find the angle θ of a right triangle with opposite side 3 and adjacent side 4.
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 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 = 3 and A = 4.
3 divided by 4 is 0.75.
4 Turn it into the angle
The tan⁻¹ button on the calculator turns a ratio back into the angle.
So θ = tan⁻¹(0.75).
The calculator gives about 36.87.
To the nearest tenth of a degree, θ ≈ 36.9°.
5 Check
Let's check: tan of 36.9° on the calculator.
0.75 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.