Evaluate Tan 240 Degrees Without Calculator

Understanding the Tangent Function

The tangent function, often written as tan(θ), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side. It's defined as:

The tangent function is periodic with a period of 180 degrees, meaning tan(θ) = tan(θ + 180°n) for any integer n. This periodicity is crucial for evaluating tangent at angles outside the standard 0° to 90° range.

Calculating tan(240°)

To find tan(240°), we need to determine the reference angle and understand the behavior of the tangent function in the given quadrant. Here's how to approach it:

1. Identify the quadrant of 240°: 240° lies in the third quadrant (180° to 270°).
2. Find the reference angle: Subtract 180° from 240° to get the reference angle of 60°.
3. Determine the sign of tangent in the third quadrant: In the third quadrant, both sine and cosine are negative, so tangent (which is sine/cosine) is positive.
4. Calculate tan(60°): The exact value of tan(60°) is √3.
5. Apply the periodicity: Since tan(θ) = tan(θ + 180°), tan(240°) = tan(60°) = √3.

Therefore, tan(240°) = √3.

Step-by-Step Calculation

Let's break down the calculation of tan(240°) into clear steps:

1. Identify the angle: We're calculating tan(240°).
2. Determine the quadrant: 240° is between 180° and 270°, placing it in the third quadrant.
3. Find the reference angle: Subtract 180° from 240° to get 60°.
4. Recall tangent properties: In the third quadrant, tangent is positive because both sine and cosine are negative.
5. Calculate tan(60°): The exact value of tan(60°) is √3 (approximately 1.732).
6. Apply periodicity: Since tan(θ) = tan(θ + 180°), tan(240°) = tan(60°) = √3.

Verification

To ensure our calculation is correct, let's verify it using the unit circle:

1. The point on the unit circle at 240° has coordinates (-1/2, -√3/2).
2. The tangent of an angle θ is equal to the y-coordinate divided by the x-coordinate of the corresponding point on the unit circle.
3. Therefore, tan(240°) = (-√3/2) / (-1/2) = √3.

This confirms our earlier result that tan(240°) = √3.

Common Mistakes

When calculating tan(240°), it's easy to make the following mistakes:

• Incorrect quadrant identification: Forgetting that 240° is in the third quadrant where tangent is positive.
• Reference angle error: Calculating the reference angle incorrectly as 180° instead of 60°.
• Sign error: Assuming tangent is negative in the third quadrant because of the negative coordinates.
• Periodicity misunderstanding: Not recognizing that tan(θ) = tan(θ + 180°).

Being aware of these common pitfalls can help ensure accurate calculations.