Finding an Angle in a Right Angled Triangle
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?
(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:

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 re-arrange that into this: x = sin-1 (0.5)
And then get our by Text-Enhance" href="" in_rurl="http://i.trkjmp.com/click?v=VVM6MzA5OTg6NDpjYWxjdWxhdG9yOmE2ODVhZjliODBlNzdiNzYyZWNkNWU4OGRlYTI4MThkOnotMTMzNS0xMjcwMDg6d3d3Lm1hdGhzaXNmdW4uY29tOjA6MA"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 your calculator, try using sin and sin-1 to see what results you get!
Also
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?
(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:

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 re-arrange that into this: x = sin-1 (0.5)
And then get our by Text-Enhance" href="" in_rurl="http://i.trkjmp.com/click?v=VVM6MzA5OTg6NDpjYWxjdWxhdG9yOmE2ODVhZjliODBlNzdiNzYyZWNkNWU4OGRlYTI4MThkOnotMTMzNS0xMjcwMDg6d3d3Lm1hdGhzaXNmdW4uY29tOjA6MA"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 your calculator, try using sin and sin-1 to see what results you get!
Also