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Tan 22.5 Without A Calculator

Reviewed by Calculator Editorial Team

Calculating tan(22.5°) without a calculator requires understanding of trigonometric identities and exact values. This guide explains the exact value, calculation methods, and provides a step-by-step solution.

Exact Value of tan(22.5°)

The exact value of tan(22.5°) can be derived using trigonometric identities. The tangent of 22.5° is equal to the square root of 2 minus 1.

Formula

tan(22.5°) = √2 - 1 ≈ 0.4142

This exact value is derived from the half-angle formula for tangent:

Half-Angle Formula

tan(θ/2) = (1 - cosθ)/sinθ

When θ = 45°, we get tan(22.5°).

Calculation Methods

There are several methods to calculate tan(22.5°) without a calculator:

  1. Using the half-angle formula for tangent
  2. Using the tangent addition formula
  3. Using the identity tan(22.5°) = √2 - 1

The most straightforward method is using the identity tan(22.5°) = √2 - 1.

Step-by-Step Calculation

Here's how to calculate tan(22.5°) using the exact value method:

  1. Recall that √2 ≈ 1.4142
  2. Subtract 1 from √2: 1.4142 - 1 = 0.4142
  3. The result is tan(22.5°) ≈ 0.4142

This method provides an exact value without needing a calculator.

Worked Example

Let's calculate tan(22.5°) using the exact value method:

Example Calculation

Given: tan(22.5°) = √2 - 1

√2 ≈ 1.4142

tan(22.5°) ≈ 1.4142 - 1 = 0.4142

The exact value of tan(22.5°) is √2 - 1, which is approximately 0.4142.

Frequently Asked Questions

What is the exact value of tan(22.5°)?
The exact value of tan(22.5°) is √2 - 1.
How can I calculate tan(22.5°) without a calculator?
You can use the exact value √2 - 1 or apply the half-angle formula for tangent.
What is tan(22.5°) in decimal form?
tan(22.5°) ≈ 0.4142.
Is tan(22.5°) the same as tan(22.5 radians)?
No, tan(22.5°) refers to the angle in degrees, while tan(22.5 radians) refers to the angle in radians.