Finding an Angle in a Right Angled Triangle

Angle from Any 2 Sides

Nosotros tin find an unknown angle in a right-angled triangle, every bit long as 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 reply is to utilise Sine, Cosine or Tangent!

Simply which i to use? We have a special phrase "SOHCAHTOA" to help u.s., and we use it like this:

Step 1: observe the names of the two sides nosotros know

triangle showing Opposite, Adjacent and Hypotenuse

  • Adjacent is adjacent to the angle,
  • Opposite is contrary the bending,
  • and the longest side is the Hypotenuse.

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

  • the side Contrary the angle "ten", which is 2.five
  • the longest side, chosen the Hypotenuse, which is 5

Footstep 2: at present utilise the showtime messages of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" to find which ane of Sine, Cosine or Tangent to utilise:

SOH...

Due southine: 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 the states we demand to use Sine.

Step three: Put our values into the Sine equation:

Due southin (x) = Opposite / Hypotenuse = 2.v / 5 = 0.5

Step four: Now solve that equation!

sin(x) = 0.5

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

ten = sin-1(0.5)

And then get our calculator, key in 0.five and utilize the sin-1 button to get the answer:

10 = xxx°

And we have our answer!

But what is the significant of sin-1 … ?

Well, the Sine office "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 ...
... information technology takes the ratio "contrary/hypotenuse" and gives us an bending.

Example:

  • Sine Function: sin(30°) = 0.5
  • Inverse Sine Role: sin-one(0.v) = xxx°
calculator-sin-cos-tan On the calculator press ane of the following (depending
on your brand of reckoner): either '2ndF sin' or 'shift sin'.

On your calculator, try using sin and sin-i to run into what results you get!

Too endeavor cos and cos-1 . And tan and tan-1 .
Go on, have a try now.

Footstep By Stride

These are the 4 steps we demand to follow:

  • Pace 1 Find which two sides nosotros know – out of Opposite, Side by side and Hypotenuse.
  • Step 2 Utilise SOHCAHTOA to decide which 1 of Sine, Cosine or Tangent to use in this question.
  • Step three For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse or for Tangent calculate Opposite/Adjacent.
  • Step 4 Find the angle from your computer, using 1 of sin-ane, cos-one or tan-i

Examples

Let's look at a couple more examples:

trig example airplane 400, 300

Case

Detect the angle of height of the aeroplane from point A on the ground.


  • Footstep 1 The two sides nosotros know are Opposite (300) and Adjacent (400).
  • Step 2 SOHCAHTOA tells us we must use Tangent.
  • Footstep 3 Calculate Opposite/Side by side = 300/400 = 0.75
  • Step 4 Find the bending from your calculator using tan-1

Tan 10° = 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.

trig example

Example

Observe the size of angle a°


  • Step 1 The two sides nosotros know are Adjacent (half-dozen,750) and Hypotenuse (8,100).
  • Pace ii SOHCAHTOA tells us we must use Cosine.
  • Step 3 Calculate Side by side / Hypotenuse = half-dozen,750/8,100 = 0.8333
  • Step four Detect the angle from your figurer using cos-1 of 0.8333:

cos a° = vi,750/eight,100 = 0.8333

cos-i of 0.8333 = 33.vi° (to ane decimal place)

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