Right triangle opposite 5, hypotenuse 10, find the angle Step by Step

Step by step

The question

Question: find the angle θ of a right triangle with opposite side 5 and hypotenuse 10.

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.

SOH CAH TOA

We have the opposite and the hypotenuse: O and H.

SOH has them.

So we use sin.

sin θ = OH

Careful!

From the other angle, the opposite and adjacent sides would swap.

Always name them from the angle we use.

Careful: O and A change with the angle

3 Write the equation

Now put the numbers in: O = 5 and H = 10.

sin θ = 510

5 divided by 10 is 0.5.

sin θ = 0.5

4 Turn it into the angle

The sin⁻¹ button on the calculator turns a ratio back into the angle.

sin⁻¹ : ratio → angle

So θ = sin⁻¹(0.5).

θ = sin⁻¹(0.5)

The calculator gives exactly 30.

So θ = 30°.

θ = 30°

5 Check

Let's check: sin of 30° on the calculator.

sin 30° = 0.5

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.

Answer