Solve Tan 4pi 3 Trig Without Calculator

Understanding tan(4π/3)

The tangent function, tan(θ), is defined as the ratio of sine to cosine for any angle θ. For tan(4π/3), we need to determine the sine and cosine values at this angle.

The angle 4π/3 radians is equivalent to 240 degrees. This places the angle in the third quadrant of the unit circle, where both sine and cosine values are negative.

Step-by-step Solution

1. Identify the Quadrant: 4π/3 radians is in the third quadrant (between π and 3π/2).
2. Find the Reference Angle: The reference angle is calculated as θ - π = 4π/3 - π = π/3 (60 degrees).
3. Determine Sine and Cosine: In the third quadrant, both sine and cosine are negative. Using the reference angle:
   • sin(π/3) = √3/2 → sin(4π/3) = -√3/2
   • cos(π/3) = 1/2 → cos(4π/3) = -1/2
4. Calculate Tangent: tan(4π/3) = sin(4π/3)/cos(4π/3) = (-√3/2)/(-1/2) = √3

Verification

To verify our solution, we can use the tangent addition formula:

Let's use A = π and B = π/3:

1. tan(π) = 0
2. tan(π/3) = √3
3. tan(π + π/3) = (0 + √3)/(1 - 0*√3) = √3

This confirms our earlier result that tan(4π/3) = √3.

Common Mistakes

• Sign Errors: Forgetting that both sine and cosine are negative in the third quadrant can lead to incorrect results.
• Reference Angle Confusion: Using the wrong reference angle (e.g., π/3 instead of 2π/3) would give incorrect values.
• Cancellation Errors: Not recognizing that the negatives in the numerator and denominator cancel out can result in -√3 instead of √3.