Finding an Angle in a Right Angled Triangle

You can find the Angle from Any Two Sides

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

Example

A 5ft ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

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 know
Step 2: Now use the initials of the sides and the phrase SOHCAHTOA" to find out what to choose among sin, cos or tan.
Step 3: Put our values into the Sine equation:
Sin (x) = Opposite / Hypotenuse = 2.5 / 5 = 0.
Step 4: Now solve that equation!
sin (x) = 0.5
Step 5:Next we can re-arrange that into this:
x = sin^-1 (0.5)
Step 6:Then get our calculator, key in 0.5 and use the sin^-1 button to get the answer:

x = 30°

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

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

• the side Opposite the angle "x" (2.5 ft)
• the long sloping side, called the “Hypotenuse” (5 ft)

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.

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 Function: sin(30°) = 0.5
• Inverse Sine Function: 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 orOpposite/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 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 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)