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Edunet site of the week January 2001

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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 (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:

sin²θ + 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

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 means you can use plus or minus, and the 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)