Syntax. TAN(number)
Definition. This function returns the tangent of an angle.
Argument
number (required) The angle in radians for which you want the tangent
Background. In a right triangle, the ratio between the opposite leg and the adjacent leg depends on the apex angle. The ratio of the opposite leg of an angle and the adjacent leg is the tangent (tan) of an angle.
The TAN() function requires radians for calculations. If the angle is specified in degrees, you have to multiply the angle by PI()/180 or convert it with the RADIANS() function.
If angle α in an unit circle (c = 1) increases between 0° and 90°, the tangent increases between 0 and ∞ (infinity) (see Figure 16-38).
In a coordinate system, the angle α on the x-axis and the tangent of α on the y-axis result in the curve shown in Figure 16-39.
The ratio between the tangent and the sine and cosine is:
The tangent indicates the correlation between the gradient angle and the slope of a straight line. For example, if the gradient angle of a street is 12°, its slope is tan(12°) and approximately 0.21. The traffic sign indicating the slope of the street reads “21%” (an elevation difference of 21 meters per 100 meters).
For a perpendicular, the slope isn’t useful because tan(90°) and tan(–90°) are not defined.
Example. Suppose you want to calculate the height of a chimney. The chimney is 130 meters away, and the angle of elevation from its top is α=24°50’. How high is the chimney if the eye level is 1.6 meters?
The formula for the tangent is:
You enter the existing information in the formula
The equation to calculate the height is
height = 130 • tan(24.5)
The result is rounded to two decimal places and 1.60 meters (eye level) is added.
h = 59.2444132 ≈ 59.24 m +1.60 m (eye level) = 60.84 m
Enter the following formula in the formula bar:
=ROUND(TAN(RADIANS(24.5)) • 130;2)+1.6
For Excel to be able to calculate the tangent for the angle of 19.5°, the angle has to be converted to radians. Then the tangent is multiplied by the opposite leg. The result is rounded to two decimal places and 1.60 meters (eye level) is added.