Tangent Without Calculator

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 trigonometry, the tangent function is one of the three primary functions (along with sine and cosine) and is often abbreviated as tan.

For any angle θ, the tangent function can be extended beyond the right-angled triangle definition using the unit circle or infinite series. The tangent function is periodic with a period of π (180 degrees) and has vertical asymptotes where the cosine function equals zero.

Methods Without Calculator

Using Right-Angled Triangles

For angles between 0 and π/2 (0° and 90°), you can construct a right-angled triangle with the given angle and measure the sides. The tangent is then the ratio of the opposite side to the adjacent side.

Using Known Values

For common angles like 30°, 45°, and 60°, you can recall the exact tangent values:

• tan(30°) = √3/3 ≈ 0.577
• tan(45°) = 1
• tan(60°) = √3 ≈ 1.732

Using Series Expansion

For angles near 0, you can use the Taylor series expansion of the tangent function:

This approximation is accurate for small angles where θ is in radians.

Step-by-Step Examples

Example 1: Using a Right-Angled Triangle

Find tan(36.87°) using a right-angled triangle with opposite side 5 and adjacent side 7.

1. Draw a right-angled triangle with angle θ = 36.87°.
2. Measure the opposite side as 5 units and the adjacent side as 7 units.
3. Calculate tan(θ) = opposite/adjacent = 5/7 ≈ 0.714.

Example 2: Using Series Expansion

Find tan(0.1 radians) using the series expansion.

1. Convert 0.1 radians to degrees: 0.1 × (180/π) ≈ 5.73°.
2. Use the first term of the series: tan(θ) ≈ θ = 0.1.
3. The result is tan(0.1) ≈ 0.1003 (using more terms would give a more precise result).

Common Angles

Here are the tangent values for common angles: