Trigonometry

Trig functions

The unit circle is a circle with radius 1 centred at the origin (possibly NIS but a useful model regardless).

For an angle drawn anticlockwise from the axis:

and , so .
and , so .

There are other trig functions:

Graphs

Graph of

The graph is even (symmetric about ) and continues indefinitely with a period of .

Graph of

The graph is odd (rotational symmetry order 2 about the origin) and also continues indefinitely with a period of .

Graph of

, so the graph passes through the origin. Both and are positive in the first quadrant, so is positive for . As gets closer to 0 as approaches , gets larger and larger, with an asymptote at . The graph continues indefinitely with a period of .

Trig identities

The two most basic trigonometric identities are:

Dividing the first identity through by or gives:

Sine rule

The sine rule

When using the sine rule to find angles, sometimes there may be two possible answers, and .

Cosine rule

Area of a triangle

The area of a triangle with sides and and an angle between them is given by:

Radians

Angles can be measured in degrees or radians, with radians in a circle (). Thus, to convert between them:

To convert from radians to degrees, multiply by
To convert from degrees to radians, multiply by .

Radians and geometry

The length of an arc of a circle with radius over angle subtended at the centre, in radians, is given by:

The area of a sector of circle with radius over angle subtended at the centre, in radians, is given by:

Small angle approximations

The small angle approximations can be used for small angles measured in radians:

Compound angle formulae

Geometric derivation

The main derivation for compound angle formulae is for , with the other formulae following from this one. IS: geometric proof of the compound-angle formulae.

Consider as below, with as an altitude to side .

The area of can be calculated:

The area of can also be calculated:

By considering the area of as a whole:

Giving the final result.

This result can be manipulated to derive the 3 other compound-angle identities for and , which we leave a proof for below. These identities can be further manipulated to derive 2 additional compound-angle identities for . The below 6 identities are given:

Phase shift form

An expression of the form can also be written in the forms:

for suitably chosen and . represents the amplitude of the resulting expression, while is a phase shift.

To rewrite an expression into these forms:

Start with the target form and apply the compound-angle identity
Compare coefficients on the and terms
The amplitude is always given by
By dividing by , and thus can be found, taking care with signs
Write the final expression with substituted values for and .

Proofs

The cosine rule

Consider such that , , and . By SAS congruency, all triangles with these properties must be congruent, so can only have one length.

is an altitude drawn to . The length of can be calculated using basic trigonometry in right angle triangle . , so .

With known, can be calculated using Pythagoras.

With and known, can be calculated using Pythagoras.

Or alternatively:

The sine rule

Here, we know and and the side between them. We draw in the circumcircle of the triangle, making a diameter of the circumcircle, which has radius .

Because and subtend the same arc, they are equal.

As there is nothing special about side , we can use the symmetry to write:

Sine and cosine compound angle formulae

Starting with $, we can apply some manipulations to find the three other variations.

To find :

Where on the third line we apply and .

To find :

Where on the third line we apply and

To find :

Giving us all four addition formulae for and :

Tangent compound angle formulae

From the and compound angle formulae, we can derive the compound angle formulae for :

Where on the third line we divide the numerator and denominator by .

Similarly: