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Trigonometric Identities

You might like to read our page on Trigonometry first!

 

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.

Right Triangle

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)

But Wait ... There is More!

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

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 () plus the square of b () is equal to the square of c ():

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 (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

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)