Calculate Tangent to Degrees by Hand

Calculating the tangent of an angle in degrees by hand requires understanding the tangent function and using trigonometric identities. This guide provides step-by-step instructions, formulas, and practical examples to help you master this essential trigonometric calculation.

What is Tangent?

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. In other words, if you have a right-angled triangle with angle θ, the tangent of θ (tan θ) is calculated as:

tan θ = opposite / adjacent

For angles outside the range of a right-angled triangle (0° to 90°), the tangent function can be extended using the unit circle and trigonometric identities. The tangent function is periodic with a period of 180°, meaning tan θ = tan (θ + 180°n) for any integer n.

Tangent Formula

The primary formula for calculating the tangent of an angle θ in degrees is:

tan θ = sin θ / cos θ

Where sin θ is the sine of the angle and cos θ is the cosine of the angle. This formula is derived from the definition of tangent in a right-angled triangle and can be used for any angle.

Note: For angles outside the range of -90° to 90°, you may need to use reference angles and the periodicity of the tangent function to find the correct value.

How to Calculate Tangent

Calculating the tangent of an angle in degrees by hand involves several steps:

Convert the angle to radians if necessary (though most calculators work directly with degrees).
Use the tangent formula: tan θ = sin θ / cos θ.
If the angle is not in the standard range (0° to 90°), use trigonometric identities to find an equivalent angle within this range.
Calculate the sine and cosine of the angle using known values or series expansions.
Divide the sine value by the cosine value to get the tangent.

For example, to calculate tan 45°:

tan 45° = sin 45° / cos 45° = (√2/2) / (√2/2) = 1

Tangent Values Table

The following table provides tangent values for common angles in degrees:

Angle (θ)	tan θ
0°	0
30°	√3/3 ≈ 0.577
45°	1
60°	√3 ≈ 1.732
90°	Undefined (infinite)

Common Tangent Angles

Here are the tangent values for some common angles:

tan 0° = 0
tan 30° ≈ 0.577
tan 45° = 1
tan 60° ≈ 1.732
tan 90° = Undefined

These values are derived from the properties of special right-angled triangles and the unit circle.

FAQ

What is the tangent of 0 degrees?

The tangent of 0 degrees is 0 because the opposite side of a right-angled triangle with a 0° angle is 0, and the adjacent side is the hypotenuse.

How do you calculate the tangent of an angle greater than 90 degrees?

For angles greater than 90 degrees, you can use the periodicity of the tangent function. The tangent of an angle θ is equal to the tangent of θ minus 180° if θ is between 90° and 270°.

What is the tangent of 90 degrees?

The tangent of 90 degrees is undefined because the cosine of 90 degrees is 0, and division by zero is not defined in mathematics.