## Trigonometric Identities

Table of Contents

## 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

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

Cosecant Function | csc(θ) = Hypotenuse / Opposite |

Secant Function | sec(θ) = Hypotenuse / Adjacent |

Cotangent Function | cot(θ) = Adjacent / Opposite |

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

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:

*a*^{2} + *b*^{2} = c^{2}

Dividing through by c^{2} 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:

**sin**^{2}** θ + cos**^{2}** θ = 1**

Note:sin

^{2}θ means to find the sine of θ, then square the result, andsin θ

^{2}means to square θ, then do the sine function

## Related identities

Related identities include

sin^{2} θ = 1 − cos^{2} θ

cos^{2} θ = 1 − sin^{2} θ

tan^{2} θ + 1 = sec^{2} θ

tan^{2} θ = sec^{2} θ − 1

cot^{2} θ + 1 = csc^{2} θ

cot^{2} θ = csc^{2} θ − 1

## 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

Sine | sin (A+B) = sin A cos B + cos A sin B sin (A-B) = sin A cos B – cos A sin B |

Cosine | cos (A+B) = cos A cos B – sin A sin B cos (A-B) = cos A cos B + sin A sin B |

Tangent | |

Cotangent |