Finding an Angle in a Right Angled Triangle

Bending from Whatever Two Sides

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

ladder against wall

Case

The ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

The respond is to use Sine, Cosine or Tangent!

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

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

triangle showing Opposite, Adjacent and Hypotenuse

  • Side by side 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.v
  • the longest side, chosen the Hypotenuse, which is five

Step 2: now utilise the showtime letters of those ii sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" to find which i of Sine, Cosine or Tangent to use:

SOH...

Southine: sin(θ) = Opposite / Hypotenuse

...CAH...

Cosine: cos(θ) = Adjacent / Hypotenuse

...TOA

Tangent: tan(θ) = Opposite / Adjacent

In our case that is Opposite and Hypotenuse, and that gives u.s.a. "SOHcahtoa", which tells usa nosotros demand to utilise Sine.

Pace three: 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.v)

And so get our computer, key in 0.5 and apply the sin-one push button to get the answer:

x = thirty°

And we have our answer!

But what is the meaning of sin-one … ?

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

sin vs sin-1

Only sin-1 (chosen "inverse sine") goes the other way ...
... it takes the ratio "opposite/hypotenuse" and gives u.s. an angle.

Example:

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

On your calculator, endeavour using sin and sin-one to see what results you lot get!

Also attempt cos and cos-1 . And tan and tan-1 .
Keep, have a try at present.

Pace By Footstep

These are the four steps we need to follow:

  • Footstep i Find which ii sides we know – out of Opposite, Adjacent and Hypotenuse.
  • Step 2 Employ SOHCAHTOA to decide which i of Sine, Cosine or Tangent to use in this question.
  • Footstep three For Sine calculate Reverse/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse or for Tangent calculate Opposite/Next.
  • Stride 4 Detect the angle from your figurer, using 1 of sin-1, cos-1 or tan-i

Examples

Allow'southward await at a couple more examples:

trig example airplane 400, 300

Example

Find the angle of elevation of the plane from betoken A on the ground.


  • Step 1 The two sides we know are Opposite (300) and Adjacent (400).
  • Pace two SOHCAHTOA tells usa we must use Tangent.
  • Step 3 Summate Opposite/Adjacent = 300/400 = 0.75
  • Stride 4 Observe the angle from your computer 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 normally rounded to one place of decimals.

trig example

Example

Notice the size of angle a°


  • Stride 1 The two sides we know are Adjacent (vi,750) and Hypotenuse (8,100).
  • Step two SOHCAHTOA tells united states of america we must use Cosine.
  • Step three Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333
  • Footstep 4 Find the angle from your calculator using cos-one of 0.8333:

cos a° = 6,750/eight,100 = 0.8333

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

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