Finding a Side in a Right-Angled Triangle (2024)

Find a Side when we know another Side and Angle

We can find an unknown side in a right-angled triangle when we know:

one length, and
one angle (apart from the right angle).

Example: Depth to the Seabed

The ship is anchored on the seabed.

We know:

the cable length (30 m), and
the angle the cable makes with the seabed

So we should be able to find the depth!

But How?

The answer is to use Sine, Cosine or Tangent!

But Which One?

Which one of Sine, Cosine or Tangent to use?

To find out which, first we give names to the sides:

Adjacent is adjacent (next to) to the angle,
Opposite is opposite the angle,
and the longest side is the Hypotenuse.

Now, for the side we already know and the side we are trying to find, we use the first letters of their names and the phrase "SOHCAHTOA" to decide which function:

*SOH...*		Sine: sin(θ) = Opposite / Hypotenuse
*...CAH...*		Cosine: cos(θ) = Adjacent / Hypotenuse
*...TOA*		Tangent: tan(θ) = Opposite / Adjacent

Like this:

Example: Depth to the Seabed (Continued)

Find the names of the two sides we are working on:

the side we know is the Hypotenuse
the side we want to find is Opposite the angle (check for yourself that "d" is opposite the angle 39°)

Now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" which gives us "SOHcahtoa", which tells us we need to use Sine:

Sine: sin(θ) = Opposite / Hypotenuse

Now put in the values we know:

Step By Step

These are the four steps to follow:

Step 1 Find the names of the two sides we are using, one we are trying to find and one we already know, out of Opposite, Adjacent and Hypotenuse.

Step 2 Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question.

Step 3 For Sine write down Opposite/Hypotenuse, for Cosine write down Adjacent/Hypotenuse or for Tangent write down Opposite/Adjacent. One of the values is the unknown length.

Step 4 Solve using your calculator and your skills with Algebra.

Examples

Let’s look at a few more examples:

Example: find the height of the plane.

We know the distance to the plane is 1000
And the angle is 60°

What is the plane's height?

Careful! The 60° angle is at the top, so the "h" side is Adjacent to the angle!

Step 1 The two sides we are using are Adjacent (h) and Hypotenuse (1000).

Step 2 SOHCAHTOA tells us to use Cosine.

Step 3 Put our values into the Cosine equation:
cos 60° = Adjacent / Hypotenuse
= h / 1000

Step 4 Solve:

Start with:cos 60° = h/1000

Swap:h/1000 = cos 60°

Calculate cos 60°:h/1000 = 0.5

Multiply both sides by 1000:h = 0.5 x 1000

h = 500

The height of the plane = 500 meters

Example: Find the length of the side y:

Step 1 The two sides we are using are Opposite (y)
and Adjacent (7).

Step 2 SOHCAHTOA tells us to use Tangent.

Step 3 Put our values into the tangent function:
tan53°=Opposite/Adjacent
=y/7

Step 4 Solve:

Start with:tan 53° = y/7

Swap:y/7 = tan 53°

Multiply both sides by 7:y = 7 tan 53°

Calculate:y = 7 x 1.32704...

y = 9.29 to 2 decimal places

Side y = 9.29

Example: Radio Mast

There is a mast that is 70 meters high.

A wire goes to the top of the mast at an angle of 68°.

How long is the wire?

Step 1 The two sides we are using are Opposite (70) and Hypotenuse (w).

Step 2 SOHCAHTOA tells us to use Sine.

Step 3 Write down:
sin 68° = 70/w

Step 4 Solve:

The unknown length is on the bottom (the denominator) of the fraction!

So we need to follow a slightly different approach when solving:

Start with:sin 68° = 70/w

Multiply both sides by w:w × (sin 68°) = 70

Divide both sides by "sin 68°":w = 70 / (sin 68°)

Calculate:w = 70 / 0.9271...

w = 75.5 m to 1 decimal place

The length of the wire = 75.5 m

258, 1504, 1505, 1506, 1507, 2346, 2347, 2348, 3935, 248

The Law of Sines The Law of Cosines Solving Triangles Trigonometry Index Algebra Index