Step by step
The question
Question: find the angle θ of a right triangle with opposite side 7 and hypotenuse 9.
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 hypotenuse: O and H.
SOH has them.
So we use sin.
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 H = 9.
7 divided by 9 is about 0.7778.
4 Turn it into the angle
The sin⁻¹ button on the calculator turns a ratio back into the angle.
So θ = sin⁻¹(0.7778).
The calculator gives about 51.058.
To the nearest tenth of a degree, θ ≈ 51.1°.
5 Check
Let's check: sin of 51.1° on the calculator.
0.7778 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.