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Tan 240 Degrees Without Calculator

Reviewed by Calculator Editorial Team

Calculating tan 240 degrees without a calculator requires understanding of trigonometric identities and reference angles. This guide explains the method step-by-step, including how to determine the quadrant and reference angle for 240 degrees.

How to Calculate tan 240° Without a Calculator

The tangent of an angle in the unit circle can be determined using trigonometric identities. For 240 degrees, which is in the third quadrant, we can use the following steps:

  1. Identify the quadrant of the angle (240° is in the third quadrant).
  2. Find the reference angle by subtracting 180° from the angle.
  3. Determine the tangent of the reference angle.
  4. Apply the sign rule for the tangent in the third quadrant.

Key Point: In the third quadrant, both sine and cosine are negative, so tangent (which is sine/cosine) is positive.

Step-by-Step Calculation

Let's calculate tan 240° step by step:

  1. Identify the quadrant: 240° is between 180° and 270°, placing it in the third quadrant.
  2. Find the reference angle: Subtract 180° from 240°: 240° - 180° = 60°.
  3. Calculate tan of the reference angle: tan 60° = √3 ≈ 1.732.
  4. Apply the sign rule: In the third quadrant, tangent is positive, so tan 240° = tan 60° = √3.

The final result is tan 240° = √3.

The Formula

The tangent of an angle θ can be calculated using the following identity:

tan θ = sin θ / cos θ

For angles in the third quadrant, the tangent is positive because both sine and cosine are negative, and the negatives cancel out.

Worked Example

Let's verify the calculation with an example:

  1. Given angle θ = 240°.
  2. Reference angle = 240° - 180° = 60°.
  3. tan 60° = √3 ≈ 1.732.
  4. Since 240° is in the third quadrant, tan 240° = tan 60° = √3.

Final Result

tan 240° = √3 ≈ 1.732

This means the ratio of the opposite side to the adjacent side in a right triangle with an angle of 240° is √3:1.

Frequently Asked Questions

Why is tan 240° positive?
Because 240° is in the third quadrant where both sine and cosine are negative, making tangent (sine/cosine) positive.
What is the reference angle for 240°?
The reference angle is 60° (240° - 180°).
Can I use this method for other angles?
Yes, this method works for any angle by first determining its quadrant and reference angle.
Is tan 240° the same as tan 60°?
Yes, because they have the same reference angle and the same sign in their respective quadrants.