The definitions for the different functions are as follows:[list][*][justify][b]Sine[/b] - 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: [math]sin\left(\theta\right)=\frac{opposite}{hypotenuse}[/math] (where [math]\theta[/math] 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.[/justify][/*][/list][table][tr][td][math]0^\circ[/math][/td][td][math]0[/math][/td][/tr][tr][td][math]15^\circ[/math][/td][td][math]\frac{\sqrt{6}-\sqrt{2}}{4}[/math][/td][/tr][tr][td][math]30^\circ[/math][/td][td][math]\frac{1}{2}[/math][/td][/tr][tr][td][math]45^\circ[/math][/td][td][math]\frac{\sqrt{2}}{2}[/math][/td][/tr][tr][td][math]60^\circ[/math][/td][td][math]\frac{\sqrt{3}}{2}[/math][/td][/tr][tr][td][math]75^\circ[/math][/td][td][math]\frac{\sqrt{6}+\sqrt{2}}{4}[/math][/td][/tr][tr][td][math]90^\circ[/math][/td][td][math]1[/math][/td][/tr][/table][list][*][justify][b]C[/b][b]osine[/b] - 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: [math]cos\left(\theta\right)=\frac{adjacent}{hypotenuse}[/math]. 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: [math]cos\left(\theta\right)=sin\left(\frac{\pi}{2}-\theta\right)[/math] which can be used based on the sine function. A table of common values is shown below.[/justify][/*][/list][table][tr][td][math]0^\circ[/math][/td][td][math]1[/math][/td][/tr][tr][td][math]15^\circ[/math][/td][td][math]\frac{\sqrt{6}+\sqrt{2}}{4}[/math][/td][/tr][tr][td][math]30^\circ[/math][/td][td][math]\frac{\sqrt{3}}{2}[/math][/td][/tr][tr][td][math]45^\circ[/math][/td][td][math]\frac{\sqrt{2}}{2}[/math][/td][/tr][tr][td][math]60^\circ[/math][/td][td][math]\frac{1}{2}[/math][/td][/tr][tr][td][math]75^\circ[/math][/td][td][math]\frac{\sqrt{6}-\sqrt{2}}{4}[/math][/td][/tr][tr][td][math]90^\circ[/math][/td][td][math]0[/math][/td][/tr][/table][list][*][b][/b][justify][b]Tangent[/b] - 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: [math]tan\left(\theta\right)=\frac{opposite}{adjacent}[/math]. The same process of producing a graph can be once again used - this time producing a tangent wave. another equation for tangent is: [math]tan\left(\theta\right)=\frac{sin\left(\theta\right)}{cos\left(\theta\right)}[/math] which can be used based on the sine and cosine functions. A common table of values is shown below.[/justify][/*][/list][table][tr][td][math]0^\circ[/math][/td][td][math]0[/math][/td][/tr][tr][td][math]15^\circ[/math][/td][td][math]2-\sqrt{3}[/math][/td][/tr][tr][td][math]30^\circ[/math][/td][td][math]\frac{\sqrt{3}}{3}[/math][/td][/tr][tr][td][math]45^\circ[/math][/td][td][math]1[/math][/td][/tr][tr][td][math]60^\circ[/math][/td][td][math]\sqrt{3}[/math][/td][/tr][tr][td][math]75^\circ[/math][/td][td][math]2+\sqrt{3}[/math][/td][/tr][tr][td][math]90^\circ[/math][/td][td][math]undefined[/math][/td][/tr][/table][list][*][b][/b][justify][b]Arc Functions[/b] - Arcsine, arccosine and arctangent are the functions used to find the angle [math]\theta[/math] using the sides of a triangle, their graphs can be found by reflecting the non-arc functions in the y-axis (e.g. [math]sin\left(x\right)\longrightarrow sin\left(y\right)[/math]). These functions are also commonly called inverse functions.[/justify][/*][/list]Note: degrees can be converted to radians by dividing by [math]\frac{\pi}{180}[/math].

For non-right-angle triangles, there are two major trigonometric rules which are as follows:[br][list][*][b][/b][justify][b]Sine Rule[/b] - The sine rule equation is: [math]\frac{a}{sin\left(A\right)}=\frac{b}{sin\left(B\right)}=\frac{c}{sin\left(C\right)}=2R[/math] (where [math]A[/math], [math]B[/math] and [math]C[/math] are the angles opposite the sides [math]a[/math], [math]b[/math] and [math]c[/math]). This equation can be rearranged to find a side or an angle as shown here: [math]a=\frac{sin\left(A\right)b}{sin\left(B\right)}[/math] (side) [math]A=sin^{-1}\left(\frac{sin\left(B\right)a}{b}\right)[/math] (angle).[/justify][/*][*][justify][b]Cosine Rule[/b] - The cosine rule equation is: [math]c^2=a^2+b^2-2ab\cdot cos\left(C\right)[/math]. This equation can be rearranged to find a side or angle as shown here: [math]c=\sqrt{a^2+b^2-2ab\cdot cos\left(C\right)}[/math] (side) [math]C=cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)[/math] (angle).[/justify][/*][/list]There are also two other minor trigonometric rules which are either unnecessary or not required for GCSE.