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How to Calculate Tan 75 Degrees

Reviewed by Calculator Editorial Team

Calculating tan 75 degrees requires understanding trigonometric identities and angle relationships. This guide explains the formula, provides a step-by-step method, includes a calculator, and answers common questions.

Introduction

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For 75 degrees, which is not one of the standard angles (30°, 45°, 60°, 90°), we need to use trigonometric identities to find its exact value.

75 degrees can be expressed as the sum of 45° and 30°, allowing us to use the tangent addition formula:

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

This formula will help us calculate tan 75° using the known values of tan 45° and tan 30°.

Formula

The exact value of tan 75° can be calculated using the tangent addition formula:

tan(75°) = tan(45° + 30°) = (tan 45° + tan 30°) / (1 - tan 45° tan 30°)

We know that:

  • tan 45° = 1
  • tan 30° = √3/3 ≈ 0.577

Substituting these values into the formula gives us the exact value of tan 75°.

Step-by-Step Calculation

  1. Identify that 75° = 45° + 30°
  2. Recall the tangent addition formula: tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  3. Substitute A = 45° and B = 30° into the formula
  4. Calculate tan 45° + tan 30° = 1 + √3/3
  5. Calculate 1 - tan 45° tan 30° = 1 - (1 × √3/3) = 1 - √3/3
  6. Divide the results from steps 4 and 5: (1 + √3/3) / (1 - √3/3)
  7. Simplify the expression to get the exact value of tan 75°

Note: The exact value of tan 75° is (1 + √3/3) / (1 - √3/3). For practical purposes, this can be simplified to 2 + √3.

Worked Example

Let's calculate tan 75° using the formula:

tan(75°) = (1 + √3/3) / (1 - √3/3)

First, calculate the numerator and denominator separately:

  • Numerator: 1 + √3/3 ≈ 1 + 0.577 ≈ 1.577
  • Denominator: 1 - √3/3 ≈ 1 - 0.577 ≈ 0.423

Now divide the numerator by the denominator:

1.577 / 0.423 ≈ 3.728

This matches the simplified exact value of 2 + √3 ≈ 3.732.

FAQ

What is the exact value of tan 75°?
The exact value is (1 + √3/3) / (1 - √3/3), which simplifies to 2 + √3.
How do I calculate tan 75° using a calculator?
Most scientific calculators have a tangent function. Enter 75 and press the tan button to get the result.
Can I use the tangent addition formula for other angles?
Yes, the tangent addition formula works for any two angles A and B where A + B is the angle you want to find the tangent of.
What is the approximate value of tan 75°?
The approximate value is 3.732, which is 2 + √3.
Is tan 75° the same as tan 105°?
No, tan 105° is negative because 105° is in the second quadrant where tangent is negative.