Item 26 Evaluate Tan240 Without Using A Calculator

Evaluating trigonometric functions like tan(240°) without a calculator requires understanding of the unit circle and reference angles. This guide explains how to determine the exact value of tan(240°) using fundamental trigonometric identities and properties.

Introduction

The tangent function, tan(θ), is periodic with a period of 180°, meaning tan(θ) = tan(θ + 180°). This property allows us to evaluate tan(240°) by finding an equivalent angle within the first 180°.

240° is located in the third quadrant of the unit circle, where both sine and cosine values are negative. The reference angle for 240° is calculated as 240° - 180° = 60°. We'll use this reference angle to find tan(240°).

Method for Evaluating tan(240°)

To evaluate tan(240°) without a calculator, follow these steps:

Identify the quadrant of 240° (third quadrant).
Find the reference angle: 240° - 180° = 60°.
Recall that tan(θ) = sin(θ)/cos(θ).
Determine the signs of sin(240°) and cos(240°) based on the quadrant.
Calculate tan(240°) using the reference angle.

Formula: tan(240°) = tan(240° - 180°) = tan(60°)

Since both 240° and 60° are in the same quadrant (third), the tangent values are equal.

Step-by-Step Calculation

Let's break down the calculation:

First, recognize that 240° is in the third quadrant where tangent is positive (since both sine and cosine are negative, and negative divided by negative is positive).
Find the reference angle: 240° - 180° = 60°.
We know that tan(60°) = √3.
Therefore, tan(240°) = tan(60°) = √3.

Note: The tangent function has a period of 180°, so tan(240°) = tan(60°). This is because the tangent function repeats every 180°.

Verification

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

At 240°, the coordinates on the unit circle are (-√3/2, -1/2).
tan(240°) = y-coordinate / x-coordinate = (-1/2) / (-√3/2) = (1/2) / (√3/2) = 1/√3 = √3/3.

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

Conclusion

By understanding the properties of the tangent function and the unit circle, we can evaluate tan(240°) without a calculator. The key steps involve identifying the reference angle and applying the periodicity of the tangent function.

Remember that tan(240°) = √3, and this value is useful in various mathematical and scientific applications.

Frequently Asked Questions

Why is tan(240°) equal to √3?: Because 240° is 60° beyond 180°, and tan(θ) = tan(θ - 180°). The tangent of 60° is √3, so tan(240°) is also √3.
Can I use a calculator to verify tan(240°)?: Yes, most calculators will confirm that tan(240°) ≈ 1.73205, which is approximately √3 (1.73205).
What is the reference angle for 240°?: The reference angle is 240° - 180° = 60°. This is the acute angle that shares the same trigonometric values.
Is tan(240°) positive or negative?: tan(240°) is positive because both sine and cosine are negative in the third quadrant, and negative divided by negative is positive.
How does the periodicity of tangent help here?: The tangent function has a period of 180°, so tan(240°) = tan(60°). This property allows us to evaluate the function at a more familiar angle.