Finding a Side in a Right Angled Triangle

You can find a Side if you know another Side and Angle

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

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

Example

Find the height of the plane.

We know one length (1000) and one angle (60°), so we should be able to solve it, but how?

The answer is to use Sine, Cosine or Tangent!

But which one to use? We have a special phrase "SOHCAHTOA" to help us, and we use it like this:

Step 1: find the names of the two sides you are working on: the side you already know, and the side you 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 one we know is the Hypotenuse
  • the one we are trying to find is Adjacent to the angle (check for yourself that "h" is adjacent to the angle 60°)

Step 2: now use the first letters of those two sides (Adjacent 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 Adjacent and Hypotenuse, and that gives us “sohCAHtoa”, which tells us we need to use Cosine.

Step 3: Put our values into the Cosine equation:

cos 60° = Adjacent / Hypotenuse = h / 1000

Step 4: Now solve that equation!

But how do we calculate "cos 60°" ... ?

calculator-sin-cos-tan

You use your calculator! type in 60 and then use the "cos" key. That's easy!

cos 60° = 0.5 (by my calculator)

So now we can put "0.5" instead of "cos 60°":

0.5 = h / 1000

Now all that is left is to rearrange it a little bit:

Start with:   0.5 = h / 1000
Swap sides:   h / 1000 = 0.5
Multiply both sides by 1000:   h = 0.5 x 1000 = 500
     

The height of the plane = 500 meters

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 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

The angle the cable makes with the seabed is 39° and the cable's length is 30 meters.

Find the depth "d" that the anchor ring lies beneath the hole in the ship’s side.

 

  • Step 1 The two sides we are using are Opposite (d) and Hypotenuse (30).
  • Step 2 SOHCAHTOA tells us we must use Sine.
  • Step 3 Write down the fraction for sin 39° = opposite/hypotenuse = d/30
  • Step 4 Solve:
Start with:   sin 39° = d/30
Swap:   d/30 = sin 39°
Calculate sin 39°:   d/30 = 0.6293…
Multiply both sides by 30:   d = 0.6293… x 30 = 18.88 to 2 decimal places.
     

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

 

Example

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 (x).
  • 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... = 75.5 m (to 1 place)
     

The length of the wire = 75.5 m