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 rightangled 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?
(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!
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

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)
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 (x) = Opposite / Hypotenuse = 2.5 / 5 = 0.5
Step 4: Now solve that equation!
sin (x) = 0.5
Next (trust me for the moment) we can rearrange that into this:
x = sin^{1} (0.5)
And then get our calculator, key in 0.5 and use the sin^{1} button to get the answer:
x = 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 sin^{1} (called "inverse sine") goes the other way ...
... it
takes the ratio “opposite/hypotenuse” and gives us an angle.
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!
Also try cos and cos^{1}. And tan and tan^{1}. Go on, have a try now.
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:
ExampleFind the size of the angle of elevation

 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.
ExampleFind 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