Trigonometric Identities

You might like to read our page on Trigonometry first!

Right Triangle

The Trigonometric Identities are equations that are true for Right Angled Triangles ...

... if it is not a Right Angled Triangle refer to our Triangle Identities page.

Each side of a right triangle has a name:

triangle showing Opposite, Adjacent and Hypotenuse
(Adjacent is adjacent to the angle, and Opposite is opposite ... of course!)

Important: We are soon going to be playing with all sorts of functions and it can get quite complex, but remember it all comes back to that simple triangle with:

  • Angle θ
  • Hypotenuse
  • Adjacent
  • Opposite

Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.

They are just the length of one side divided by another

For a right triangle with an angle θ :

Right-Angled Triangle
Sine Function:
 sin(θ) = Opposite / Hypotenuse
Cosine Function:
 cos(θ) = Adjacent / Hypotenuse
Tangent Function:
 tan(θ) = Opposite / Adjacent

Also, if we divide Sine by Cosine we get:

So we can also say:

tan(θ) = sin(θ)/cos(θ)

That is our first Trigonometric Identity.

But Wait ... There is More!

We can also divide "the other way around" (such as Adjacent/Opposite instead of Opposite/Adjacent):

Right-Angled Triangle
Cosecant Function:
 csc(θ) = Hypotenuse / Opposite
Secant Function:
 sec(θ) = Hypotenuse / Adjacent
Cotangent Function:
 cot(θ) = Adjacent / Opposite

Example: if Opposite = 2 and Hypotenuse = 4 then

sin(θ) = 2/4, and csc(θ) = 4/2

Because of all that we can say:

sin(θ) = 1/csc(θ) cos(θ) = 1/sec(θ) tan(θ) = 1/cot(θ)

And the other way around:

csc(θ) = 1/sin(θ) sec(θ) = 1/cos(θ) cot(θ) = 1/tan(θ)

And we also have:

cot(θ) = cos(θ)/sin(θ)

Pythagoras Theorem

For the next trigonometric identities we start with Pythagoras' Theorem:

right angled triangle

The Pythagorean Theorem states that, in a right triangle,the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2

Dividing through by c2 gives

a2 + b2 = c2
c2 c2 c2

This can be simplified to:

(a/c)2 + (b/c)2 = 1

Now, a/c is Opposite / Hypotenuse, which is sin(θ)

And b/c is Adjacent / Hypotenuse, which is cos(θ)

So (a/c)2 + (b/c)2 = 1 can also be written:

sin2 θ + cos2 θ = 1

Note: writing sin2 θ means to find the sine of θ, then square it.

If I had written sin θ2 I would have meant "square θ, then do the sine function"


Example: when the angle θ is 1 radian (approximately 57°):


sin(θ) = 84.1/100 = 0.841
cos(θ) = 54.0/100 = 0.540

0.8412 + 0.5402 = 0.707 + 0.292 = 0.999

(Close enough to 1, considering we only used 3 decimal places)

Related identities include:

  • sin2θ = 1 − cos2θ
  • cos2θ = 1 − sin2θ
  • tan2θ + 1 = sec2θ
  • tan2θ = sec2θ − 1
  • 1 + cot2θ = csc2θ
  • cot2θ = csc2θ − 1

 

How Do You Remember Them?

The identities mentioned so far can be remembered
using one clever diagram called the Magic Hexagon:

 

  

More Identitites

There are many more identities ... here are some of the more useful ones:

Opposite Angle Identities

sin (-θ) = - sin (θ) cos (-θ) = cos (θ) tan (-θ) = - tan (θ)

 

Double Angle Identities

  sin 2a
   
  cos 2a
   
  tan 2a

 

Half Angle Identities

Note that "±" means it may be either one, depending on the value of θ/2

  sin a/2
   
  cos a/2
   
  tan a/2
   
  cot a/2

Angle Sum and Difference Identities

Note that plus/minus means you can use plus or minus, and the minus/plus means to use the opposite sign.

Sum and difference identities

Triangle Identities

There are also Triangle Identities which apply to all triangles (not just Right Angled Triangles)

 

 
Sine, Cosine and Tangent Unit Circle