Worked examples
The question
Question: find the opposite side x of a right triangle with angle 30° and hypotenuse 12.
1 Name the sides
We look from the angle 30°.
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 hypotenuse and we want the opposite: H and O.
SOH has them.
So we use sin.
sin 30° = 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: H = 12 and O = x.
sin 30° = x12
4 Find x
x is on top, over 12.
To remove the 12 at the bottom, we multiply by 12.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
12 × sin 30° = 12 × x12
12/12 is 1.
So just x is left.
So x = 12 × sin 30°.
x = 12 × sin 30°
sin 30° is exactly 0.5.
That one is good to know.
sin 30° = 0.5
Put the number in.
x = 12 × 0.5
That is 6.
So x = 6.
x = 6
5 Check
Let's check: put x back in O over H.
We get 0.5.
612 = 0.5
That is sin 30° exactly.
So x is right!
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: find x.
Last: check.
▶ Watch it solved
The question
Question: find the hypotenuse x of a right triangle with angle 35° and adjacent side 8 cm.
1 Name the sides
We look from the angle 35°.
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 adjacent and we want the hypotenuse: A and H.
CAH has them.
So we use cos.
cos 35° = AH
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: A = 8 cm and H = x.
cos 35° = 8x
4 Find x
x is at the bottom.
First we multiply by x.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
x × cos 35° = x × 8x
x/x is 1.
Only the top is left.
Now we have x × cos 35° = 8.
x × cos 35° = 8
To remove times cos 35°, we divide by it.
We do it on both sides, so they stay equal.
x × cos 35°cos 35° = 8cos 35°
It divides by itself: 1.
So just x is left.
The calculator gives cos 35°: about 0.8192.
cos 35° ≈ 0.8192
Put the number in.
x ≈ 80.8192
8 divided by 0.8192 is about 9.766, with the calculator.
To the nearest tenth, x ≈ 9.8 cm.
x ≈ 9.8 cm
5 Check
Let's check: put x back in A over H.
We get 0.8163.
89.8 = 0.8163
cos 35° is about 0.8192.
Very close: the small gap is only from rounding.
So x is right!
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: find x.
Last: check.
▶ Watch it solved
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.
▶ Watch it solved
