Finding an Angle in a Right Angled Triangle

Angle from Whatsoever Two Sides

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

ladder against wall

Example

The ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

The respond is to apply Sine, Cosine or Tangent!

Just which one to use? We accept a special phrase "SOHCAHTOA" to help united states, and we use information technology like this:

Stride 1: notice the names of the 2 sides we know

triangle showing Opposite, Adjacent and Hypotenuse

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

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

  • the side Opposite the angle "10", which is two.5
  • the longest side, called the Hypotenuse, which is 5

Footstep 2: now use the start letters of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" to detect which one of Sine, Cosine or Tangent to utilize:

SOH...

Sine: sin(θ) = Opposite / Hypotenuse

...CAH...

Cosine: cos(θ) = Adjacent / Hypotenuse

...TOA

Tangent: tan(θ) = Opposite / Adjacent

In our instance that is Opposite and Hypotenuse, and that gives us "SOHcahtoa", which tells united states nosotros need to use Sine.

Step three: Put our values into the Sine equation:

Sin (10) = Opposite / Hypotenuse = 2.five / 5 = 0.5

Footstep four: Now solve that equation!

sin(x) = 0.5

Adjacent (trust me for the moment) nosotros can re-accommodate that into this:

x = sin-one(0.five)

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

x = 30°

And we have our answer!

But what is the significant of sin-1 … ?

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

sin vs sin-1

Just sin-one (chosen "inverse sine") goes the other mode ...
... it takes the ratio "contrary/hypotenuse" and gives us an angle.

Example:

  • Sine Function: sin(xxx°) = 0.5
  • Inverse Sine Function: sin-one(0.5) = thirty°
calculator-sin-cos-tan On the calculator 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 come across what results you lot get!

Also effort cos and cos-1 . And tan and tan-one .
Go on, take a try now.

Stride By Step

These are the four steps we need to follow:

  • Pace ane Notice which two sides we know – out of Reverse, Side by side and Hypotenuse.
  • Stride two Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question.
  • Pace 3 For Sine summate Opposite/Hypotenuse, for Cosine calculate Next/Hypotenuse or for Tangent calculate Reverse/Next.
  • Step 4 Notice the bending from your reckoner, using one of sin-1, cos-1 or tan-ane

Examples

Permit's look at a couple more than examples:

trig example airplane 400, 300

Example

Find the angle of peak of the plane from signal A on the footing.


  • Step i The two sides we know are Opposite (300) and Adjacent (400).
  • Footstep 2 SOHCAHTOA tells us nosotros must utilize Tangent.
  • Stride 3 Summate Opposite/Adjacent = 300/400 = 0.75
  • Step 4 Discover the bending 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 yous're told otherwise, angles are usually rounded to 1 place of decimals.

trig example

Example

Notice the size of angle a°


  • Step i The 2 sides we know are Adjacent (6,750) and Hypotenuse (8,100).
  • Pace ii SOHCAHTOA tells the states we must use Cosine.
  • Step three Summate Adjacent / Hypotenuse = half dozen,750/eight,100 = 0.8333
  • Step iv Find the angle from your calculator using cos-1 of 0.8333:

cos a° = six,750/8,100 = 0.8333

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

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