Trigonometric Functions

Trigonometric Function Definitions and Right-Angle Triangle Formulae

The definitions for the different functions are as follows:

Sine - The ratio of the length of the opposite side (side opposite the angle) to the hypotenuse (longest side) in a right-angle triangle. This can be calculated using the following equation: (where is the angle). By inserting known values into the equation and simplifying, we can plot a graph of the sine function - known as a sine wave - and interpolate it to predict unknown values. A table of common values is shown below.

Cosine - The ratio of the length of the adjacent side (side adjacent or next to the angle) to the hypotenuse in a right-angle triangle. This can be calculated using the following equation: . The same process as used with the sine equation can be taken to get a graph - known as a cosine wave. Another equation for cosine is: which can be used based on the sine function. A table of common values is shown below.

Tangent - The ratio of the length of the opposite side to the adjacent side in a right-angle triangle. This can be calculated using the following equation: . The same process of producing a graph can be once again used - this time producing a tangent wave. another equation for tangent is: which can be used based on the sine and cosine functions. A common table of values is shown below.

Arc Functions - Arcsine, arccosine and arctangent are the functions used to find the angle using the sides of a triangle, their graphs can be found by reflecting the non-arc functions in the y-axis (e.g. ). These functions are also commonly called inverse functions.

Note: degrees can be converted to radians by dividing by

What is the value of ?

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What is the general formula for the sine function of a right-angle triangle?

For non-right-angle triangles, there are two major trigonometric rules which are as follows:

Sine Rule - The sine rule equation is: (where , and are the angles opposite the sides , and ). This equation can be rearranged to find a side or an angle as shown here: (side) (angle).
Cosine Rule - The cosine rule equation is: . This equation can be rearranged to find a side or angle as shown here: (side) (angle).

There are also two other minor trigonometric rules which are either unnecessary or not required for GCSE.