Trigonometric Identities

Right Triangle

The Trigonometric Identities are equations that are true for Right Angled Triangles. (If it is not a Right Angled Triangle go to the Triangle Identities page.)

Each side of a right triangle has a name:

triangle showing Opposite, Adjacent and Hypotenuse

examples of Opposite, Adjacent and Hypotenuse

Adjacent is always next to the angle

And Opposite is opposite the angle

We are soon going to be playing with all sorts of functions, 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 :

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent

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

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

When we divide Sine by Cosine we get:

sin(θ)cos(θ) = Opposite/HypotenuseAdjacent/Hypotenuse = OppositeAdjacent = tan(θ)

So we can say:

tan(θ) = sin(θ)cos(θ)

That is our first Trigonometric Identity.

Cosecant, Secant and Cotangent

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

triangle showing Opposite, Adjacent and Hypotenuse

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

Example: when 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:

The Pythagorean Theorem says 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 c² + b² 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:

sin² θ means to find the sine of θ, then square the result, and
sin θ² means to square θ, then do the sine function

Example: 32°

Using 4 decimal places only:

sin(32°) = 0.5299...
cos(32°) = 0.8480...

Now let's calculate sin²θ + cos² θ:

0.5299² + 0.8480²
= 0.2808... + 0.7191...
= 0.9999...

We get very close to 1 using only 4 decimal places. Try it on your calculator, you might get better results!

Related identities include:

sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ + 1 = sec² θ
tan² θ = sec² θ − 1
cot² θ + 1 = csc² θ
cot² θ = csc² θ − 1

How Do You Remember Them?

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

But Wait ... There is More!

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

Half Angle Identities

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

sin(A plus/minus B) = sin(A)cos(B) cos(A)sin(B)

cos(A plus/minus B) = cos(A)cos(B) minus/plus sin(A)sin(B)

tan(A plus/minus B) = tan(A) tan(B)1 tan(A)tan(B)

cot(A plus/minus B) = cot(A)cot(B) 1cot(B) cot(A)

Triangle Identities

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