## Right Triangle

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

Each side of a right triangle has a name:

• Adjacent is always next to the angle
• Opposite is opposite the angle

## Sine, Cosine and Tangent

3 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 :

 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:

So we can say:

## Cosecant, Secant and Cotangent

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

Example

when Opposite = 1 and Hypotenuse = 2 then

sin(θ) = 1/2, and

csc(θ) = 2/1

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)

## 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

This can be simplified to:

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

## Related identities

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

## Opposite Angle Identities

sin(−θ) = −sin(θ)

cos(−θ) = cos(θ)

tan(−θ) = −tan(θ)

## Half Angle Identities

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

## Angle Sum and Difference Identities

 Sine sin (A+B) = sin A cos B + cos A sin Bsin (A-B) = sin A cos B – cos A sin B Cosine cos (A+B) = cos A cos B – sin A sin Bcos (A-B) = cos A cos B + sin A sin B Tangent Cotangent 