Finding an Angle in a Right Angled Triangle

Angle from Any 2 Sides

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

ladder against wall

Instance

The ladder leans confronting a wall as shown.

What is the angle between the ladder and the wall?

The respond is to utilize Sine, Cosine or Tangent!

But which one to use? We take a special phrase "SOHCAHTOA" to assist the states, and we utilise it like this:

Step i: find the names of the two sides nosotros know

triangle showing Opposite, Adjacent and Hypotenuse

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

Case: in our ladder instance nosotros know the length of:

  • the side Opposite the angle "x", which is 2.v
  • the longest side, called the Hypotenuse, which is 5

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:

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 u.s. "SOHcahtoa", which tells us we need to employ Sine.

Step 3: Put our values into the Sine equation:

Southwardin (x) = Opposite / Hypotenuse = 2.five / v = 0.v

Step iv: Now solve that equation!

sin(x) = 0.v

Next (trust me for the moment) nosotros can re-suit that into this:

x = sin-1(0.5)

And and so get our calculator, key in 0.v and utilize the sin-1 button to go the answer:

x = 30°

And we have our answer!

But what is the meaning of sin-1 … ?

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

sin vs sin-1

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

Instance:

  • Sine Function: sin(30°) = 0.v
  • Changed Sine Function: sin-1(0.5) = xxx°
calculator-sin-cos-tan On the calculator press ane 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 see what results you get!

Also try cos and cos-1 . And tan and tan-1 .
Proceed, take a endeavour now.

Step By Step

These are the four steps we need to follow:

  • Step ane Find which two sides nosotros know – out of Reverse, Adjacent and Hypotenuse.
  • Stride two Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to utilize in this question.
  • Step 3 For Sine calculate Opposite/Hypotenuse, for Cosine summate Adjacent/Hypotenuse or for Tangent summate Opposite/Adjacent.
  • Stride 4 Find the angle from your reckoner, using 1 of sin-1, cos-1 or tan-one

Examples

Let's look at a couple more examples:

trig example airplane 400, 300

Example

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


  • Step 1 The two sides we know are Opposite (300) and Adjacent (400).
  • Step 2 SOHCAHTOA tells us nosotros must use Tangent.
  • Step 3 Calculate Opposite/Next = 300/400 = 0.75
  • Stride iv Find the angle from your calculator using tan-1

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

tan-1 of 0.75 = 36.nine° (right to 1 decimal place)

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

trig example

Example

Notice the size of angle a°


  • Footstep 1 The two sides we know are Adjacent (6,750) and Hypotenuse (8,100).
  • Step two SOHCAHTOA tells us we must employ Cosine.
  • Footstep 3 Calculate Adjacent / Hypotenuse = six,750/8,100 = 0.8333
  • Step four Detect the angle from your calculator using cos-1 of 0.8333:

cos a° = 6,750/8,100 = 0.8333

cos-1 of 0.8333 = 33.half dozen° (to 1 decimal place)

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