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How to Find Tan 75 Without Calculator

Reviewed by Calculator Editorial Team

Calculating tan 75° without a calculator requires using trigonometric identities and properties of angles. This guide explains two reliable methods to find the tangent of 75 degrees using fundamental trigonometric formulas.

Method 1: Using Angle Sum Identity

The angle sum identity for tangent allows us to express tan(75°) in terms of tan(45°) and tan(30°), which are standard angles with known values.

Formula

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Step-by-Step Calculation

  1. Express 75° as the sum of 45° and 30°: 75° = 45° + 30°
  2. Apply the angle sum identity: tan(45° + 30°) = (tan 45° + tan 30°) / (1 - tan 45° tan 30°)
  3. Substitute known values: tan 45° = 1, tan 30° = √3/3
  4. Calculate numerator: 1 + √3/3 = (3 + √3)/3
  5. Calculate denominator: 1 - (1 * √3/3) = (3 - √3)/3
  6. Divide numerator by denominator: [(3 + √3)/3] / [(3 - √3)/3] = (3 + √3)/(3 - √3)
  7. Rationalize the denominator: Multiply numerator and denominator by (3 + √3)
  8. Final result: tan 75° = (3 + √3)² / (9 - 3) = (9 + 6√3 + 3)/6 = (12 + 6√3)/6 = 2 + √3

This method works because 75° is the sum of two standard angles (45° and 30°) whose tangent values are known. The rationalization step ensures the result is in simplest form.

Method 2: Using Half-Angle Formula

Another approach is to use the half-angle formula for tangent, starting from tan(150°).

Formula

tan(θ/2) = (1 - cos θ) / sin θ

Step-by-Step Calculation

  1. Express 75° as half of 150°: 75° = 150°/2
  2. Apply the half-angle formula: tan(150°/2) = (1 - cos 150°) / sin 150°
  3. Substitute known values: cos 150° = -√3/2, sin 150° = 1/2
  4. Calculate numerator: 1 - (-√3/2) = 1 + √3/2
  5. Calculate denominator: 1/2
  6. Divide numerator by denominator: (1 + √3/2) / (1/2) = 2 + √3

This method is valid because 150° is a standard angle with known sine and cosine values. The result matches the previous method, confirming consistency.

Comparison of Methods

Method Complexity Steps Required Final Result
Angle Sum Identity Moderate 8 steps 2 + √3
Half-Angle Formula Moderate 6 steps 2 + √3

Both methods yield the same result, but the half-angle approach requires fewer steps. Choose the method that feels most intuitive for your understanding of trigonometric identities.

FAQ

Why can't I just use a calculator for tan 75°?

While calculators provide quick results, understanding the underlying trigonometric identities helps in solving more complex problems and verifying calculator outputs.

Is tan 75° equal to tan 15°?

No, tan 75° = 2 + √3 ≈ 3.732, while tan 15° = 2 - √3 ≈ 0.268. The tangent function is not symmetric about 45°.

Can I use these methods for other angles?

Yes, these methods can be adapted for other angles by choosing appropriate standard angles to combine or halve.