Finding an Angle in a Right Angled Triangle

Angle from Whatsoever Two Sides

Nosotros tin find an unknown bending in a correct-angled triangle, as long equally we know the lengths of two of its sides.

ladder against wall

Example

The ladder leans against a wall as shown.

What is the bending 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 employ it like this:

Step 1: find the names of the two sides we know

triangle showing Opposite, Adjacent and Hypotenuse

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

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

  • the side Reverse the angle "x", which is 2.5
  • the longest side, chosen the Hypotenuse, which is 5

Step 2: at present use the first letters of those ii 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 apply Sine.

Step iii: Put our values into the Sine equation:

Sin (10) = Opposite / Hypotenuse = ii.5 / 5 = 0.5

Footstep iv: Now solve that equation!

sin(x) = 0.v

Side by side (trust me for the moment) we can re-conform that into this:

ten = sin-i(0.5)

And so get our calculator, key in 0.5 and utilise the sin-1 button to get the answer:

x = 30°

And we take our answer!

Simply what is the meaning of sin-i … ?

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

sin vs sin-1

But sin-1 (called "inverse sine") goes the other way ...
... it takes the ratio "contrary/hypotenuse" and gives us an angle.

Instance:

  • Sine Function: sin(30°) = 0.5
  • Inverse Sine Function: sin-1(0.5) = 30°
calculator-sin-cos-tan On the calculator printing one of the post-obit (depending
on your brand of calculator): either '2ndF sin' or 'shift sin'.

On your calculator, try using sin and sin-1 to run into what results y'all get!

Also try cos and cos-ane . And tan and tan-one .
Go along, have a attempt now.

Step By Step

These are the 4 steps we need to follow:

  • Step 1 Find which 2 sides we know – out of Reverse, Adjacent and Hypotenuse.
  • Footstep 2 Apply SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question.
  • Step iii For Sine summate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse or for Tangent calculate Opposite/Adjacent.
  • Step 4 Find the angle from your calculator, using one of sin-1, cos-i or tan-1

Examples

Let's expect at a couple more examples:

trig example airplane 400, 300

Example

Find the bending of summit of the plane from point A on the ground.


  • Stride i The two sides we know are Opposite (300) and Adjacent (400).
  • Pace two SOHCAHTOA tells usa we must use Tangent.
  • Step three Summate Opposite/Adjacent = 300/400 = 0.75
  • Pace iv Notice the angle from your reckoner using tan-1

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

tan-ane of 0.75 = 36.9° (right to 1 decimal place)

Unless you're told otherwise, angles are unremarkably rounded to one place of decimals.

trig example

Example

Observe the size of angle a°


  • Step ane The two sides nosotros know are Adjacent (6,750) and Hypotenuse (eight,100).
  • Pace ii SOHCAHTOA tells us we must use Cosine.
  • Footstep three Summate Side by side / Hypotenuse = half-dozen,750/8,100 = 0.8333
  • Stride four Observe the angle from your calculator using cos-1 of 0.8333:

cos a° = six,750/eight,100 = 0.8333

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

250, 1500, 1501, 1502, 251, 1503, 2349, 2350, 2351, 3934