Finding a Side in a Right-Angled Triangle

We can 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, that is).

Ship Anchor Triangle

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 to use? "SOHCAHTOA" can help us, and we use it like this:

Step 1: find the names of the two sides we are working on: the side we already know, and the side we are trying to find:

  • Adjacent is adjacent to the angle,
  • Opposite is opposite the angle,
  • and the longest side is the Hypotenuse.
triangle showing Opposite, Adjacent and Hypotenuse

In our example:

  • 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 60°)

Step 2: now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" to find which one of Sine, Cosine or Tangent to use:

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

In our example that is Opposite and Hypotenuse, and that gives us "SOHcahtoa", which tells us we need to use Sine.

Step 3: Put our values into the Sine equation:

sin 39° = Opposite / Hypotenuse = = d / 30

Step 4: Now solve that equation!

But how do we calculate sin 39° ... ?

calculator-sin-cos-tan

Use your calculator! Type in 39 and then use the "sin" key.
That's easy!

sin 39° = 0.6293… (by my calculator)

So now we can put 0.6293… instead of sin 39° :

0.6293… = d / 30

Now we rearrange it a little bit, and solve:

Start with:   0.6293… = d / 30
Swap sides:   d / 30 = 0.6293…
Multiply both sides by 30:   d = 0.6293… x 30
    d = 18.88 to 2 decimal places

The depth the anchor ring lies beneath the hole is 18.88 m

Step By Step

These are the four steps to follow:

  • Step 1 Decide which 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 Write down the fraction Opposite/Hypotenuse, Adjacent/Hypotenuse or Opposite/Adjacent, whichever is appropriate (one of the values will be 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 we must 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 a:

 

  • Step 1 The two sides we are using are Opposite (a)
    and Adjacent (7).
  • Step 2 SOHCAHTOA tells us we must use Tangent.
  • Step 3 Write down the fraction for tan 53° = Opposite/Adjacent = a/7
  • Step 4 Solve:
Start with:   tan 53° = a/7
Swap:   a/7 = tan 53°
Calculate tan 53°:   a/7 = 1.32704…
Multiply both sides by 7:   a = 1.32704… × 7 = 9.29 (to 2 decimal places)
     

Side "a" = 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 we must use Sine.
  • Step 3 Write down the fraction for 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 place)

The length of the wire = 75.5 m