There are three basic functions that are used in trigonometry, these are known as sine (sin), cosine (cos) and tangent (tan). These functions are used as part of the sine, cosine and tangent laws respectively, as well as the laws that describe the relationships between sides and angles in right triangles. Sine (sin) is a wave shape (Figure 1), alternating progressively between a maximum value of 1, and a minimum value of -1. At 90° the sine of the value is 1, while at 270° the sine is -1. The changes from these values from minimum, to maximum, and back to minimum are not linear, and rather follow the pattern of a rolling waveform.

Figure 1: The shape of the sine function between 0° and 360°. The function follows a waveform that varies between -1 and 1 which reaches a maximum at 90°and a minimum at 270°, which are directly opposite directions on a circle.

Cosine (cos) is another wave shape very similar to the sine wave (Figure 2), which also varies between the -1 and 1. However the cosine wave has different minimum and maximum points. Cosine is at a maximum at 0° and 360°, while it will be at a minimum at 180°. Apart from these different minimum and maximum points, the sine and cosine functions are very similar in form.

Figure 2: The shape of a cosine function between 0° and 360°. The cosine function is a waveform that varies between -1 and 1, and reaches maxima at 0° and 360° and has a minimum at 180°, which are directly opposite directions on a circle

Tangent (tan) is the third trigonometry function, and like the other two functions is a waveform (Figure 3). Tangent is unlike the other forms in that it has a asymptote, that is a value on an axis (in this case the x axis) that its function can approach but can never reach. This value is at 90°, where the result of the function is undefined. The tangent function does not have maximum or minimum values for this reason, results get higher as they increase towards 90 (Tan(89.999) = 57295 for example), but they are very small immediately after the asymptote (Tan(90.001) = -57295). This asymptote comes about because of how tangent is defined, which is Tanθ = Sinθ/Cosθ, as dividing by zero produces an undefined answer, this make the tangent of 90° = 0.

Figure 3: The shape of a tangent function between 0° and 360°. The tangent function extends beyond -1 and 1 unlike the other two trigonometric functions and has a period of 180° rather than 360°, and has asymptotes at 90° and 270° which have undefined values, and the lines of the tangent functions approach ininity and negative infinity as they get closer to these asymptotes.

Sine, cosine and tangent are all periodic functions, this mean that they repeat a pattern over a defined period (Figure 4). This period is 360°, this makes sense, as this is also the complete rotation of a circle, and angles below 0° and above 360° have equivalent angles within the 0° to 360°range.

Figure 4: All the trigonometric functions repeat over a period of 360°, thus angles outside the range of 0° to 360° can be considered equivalent to angles within that range. The tangent function has asymptotes at 180° intervals, and these coincide with when the cosine function is 0 (also at 180° intervals), as the tangent function is defined as the sine of the angle divided by the cosine of an angle, and division by 0 leads to an undefined result.

Like any other mathematical operator, sine, cosine and tangent all have inverse functions. These are expressed using the exponent of -1, thus the opposite of sin is sin^-1, the opposite of cos is cos^-1 and the opposite of tan is tan^-1. Using these operators, it is possible to turn sin, cos and tan values into angles.