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Finding an Angle in a Right Angled Triangle

You can find the Angle from Any Two Sides

We can find any unknown angle from a right-angled triangle, as long as we know the lengths of two of its sides.

A 5ft ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

(Note: we also solve this on Solving Triangles by Reflection but now we solve it in a more general way.)

The answer is to use Sine, Cosine or Tangent!

Which One?

Sine, Cosine or Tangent? Which one to use? It depends which two sides you know:

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

triangle showing Opposite, Adjacent and Hypotenuse

Then use the phrase "sohcahtoa" to decide which one of Sine, Cosine or Tangent to use:

*Soh...*	Sine = Opposite / Hypotenuse
*...cah...*	Cosine = Adjacent / Hypotenuse
*...toa*	Tangent = Opposite / Adjacent

Example: in our ladder example we know the length of:

the side opposite the angle (called the “Opposite”), and
the long sloping side (called the “hypotenuse”)

And in “sohcahtoa” that is “o” and “h”, and that gives us “SOHcahtoa” ...

... which tells us to use Sine

Then do the calculations like this:

Sine = Opposite / Hypotenuse = 2.5 / 5 = 0.5

Now we get out our calculator, key in 0.5 and use the sin^-1 button to get the answer:

30°

What is sin^-1 ?

But what is the meaning of sin^-1 … ?

Well, the Sine function "sin" takes an angle and gives us the ratio “opposite/hypotenuse”,

But in this case we know the ratio “opposite/hypotenuse” but want to know the angle.

So we want to go backwards. That is why we we use sin^-1, which means “inverse sine”.

Example:

Sine: sin(30°) = 0.5
Inverse Sine: sin^-1(0.5) = 30°

On the calculator you would press one of the following (depending on your brand of calculator): either '2ndF sin' or 'shift sin'.

On your calculator, try using "sin" and "sin^-1" to see what results you get!

Step By Step

These are the four steps we need to follow:

Step 1 Decide which two sides we 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 Use your calculator to calculate the fraction Opposite/Hypotenuse, Adjacent/Hypotenuse or Opposite/Adjacent (whichever is appropriate).
Step 4 Find the angle from your calculator, using one of sin^-1, cos^-1 or tan^-1

Examples

Let’s look at a couple more examples:

Example 1

Find the size of the angle of elevation of the plane from point A on the ground.

Step 1 The two sides we know are Opposite (300) and Adjacent (400).
Step 2 SOHCAHTOA tells us we must use Tangent.
Step 3 Use your calculator to calculate Opposite/Adjacent = 300/400 = 0.75
Step 4 Find the angle from your calculator using tan^-1

Tan x° = opposite/adjacent = 300/400 = 0.75

tan^-1 of 0.75 = 36.9° (correct to 1 decimal place)

Unless you’re told otherwise, angles are usually rounded to one place of decimals.

Example 2

Find the size of angle a°

Step 1 The two sides we know are Adjacent (6,750) and Hypotenuse (8,100).
Step 2 SOHCAHTOA tells us we must use Cosine.
Step 3 Use your calculator to calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333
Step 4 Find the angle from your calculator using cos^-1 of 0.8333:

cos a° = 6,750/8,100 = 0.8333

cos^-1 of 0.8333 = 33.6° (to 1 decimal place)

Question 1 Question 2

And you can find more trigonometry puzzles on the Random Trigonometry page.