Tan is one of the three main trigonometric functions, along with sin and cos. It is used to calculate the ratio of the opposite side and the adjacent side of a right-angled triangle, given an angle. In other words, tan(angle) = opposite/adjacent.

Tan can also be defined as the ratio of sin and cos of the same angle, that is, tan(angle) = sin(angle)/cos(angle). This formula can be useful when working with trigonometric identities or solving equations involving tan.

The graph of tan is a periodic function that repeats every 180 degrees or pi radians. It has vertical asymptotes at angles where cos is zero, such as 90 degrees or pi/2 radians. The graph of tan is symmetrical about the origin, which means that tan(-angle) = -tan(angle).

Tan can be used to find the angle of a right-angled triangle, given the lengths of the opposite and adjacent sides. This can be done by using the inverse tan function, also called arctan or tan^-1. For example, if opposite = 3 and adjacent = 4, then angle = arctan(3/4) = 36.87 degrees.

Tan can also be used to model real-world phenomena that involve periodic oscillations, such as sound waves, pendulums, or circular motion. For example, the position of a point on a unit circle can be expressed as (cos(angle), sin(angle)), and the slope of the tangent line at that point is tan(angle).

Here are some examples of how to use tan in different situations:

• To find the height of a building, given the distance from the base and the angle of elevation: height = distance × tan(angle)
• To find the angle of depression, given the height of a cliff and the distance to the horizon: angle = arctan(height/distance)
• To find the period of a pendulum, given the length and the gravitational acceleration: period = 2π × sqrt(length/gravity) × (1 + (tan(angle/4))^2)

For more information on tan and other trigonometric functions, you can visit these websites:

1. Sin, Cos and Tan – Maths GCSE Revision
2. Tangent calculator | tan(x) calculator – RapidTables
3. tan(x) | tangent function – RapidTables

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Tan is related to the other trigonometric functions by various identities and formulas. For example, the Pythagorean identity states that sin^2(angle) + cos^2(angle) = 1, which can be rearranged to get 1 + tan^2(angle) = sec^2(angle), where sec is the reciprocal of cos. Another identity is the double angle formula, which says that tan(2 × angle) = 2 × tan(angle) / (1 – tan^2(angle)). These identities can help simplify expressions or solve equations involving tan.

Tan can also be used to find the area of a triangle, given two sides and the angle between them. This can be done by using the formula area = (side1 × side2 × sin(angle)) / 2, and then substituting sin(angle) with tan(angle) × cos(angle). For example, if side1 = 5, side2 = 7, and angle = 30 degrees, then area = (5 × 7 × sin(30)) / 2 = (5 × 7 × (tan(30) × cos(30))) / 2 = 8.75 square units.

Tan can also be used to find the length of an arc or the area of a sector of a circle, given the radius and the central angle. This can be done by using the formulas arc = radius × angle and sector = (radius^2 × angle) / 2, and then converting the angle from degrees to radians by multiplying it by pi/180. For example, if radius = 10 and angle = 60 degrees, then arc = 10 × (60 × pi/180) = 10.47 units and sector = (10^2 × (60 × pi/180)) / 2 = 52.36 square units.