Tan 180 Without Calculator

Understanding tan(180)

The tangent function, tan(θ), is defined as the ratio of the sine of an angle to the cosine of that angle:

For θ = 180°, we need to evaluate sin(180°) and cos(180°).

From the unit circle:

• sin(180°) = 0
• cos(180°) = -1

Therefore, tan(180°) = sin(180°) / cos(180°) = 0 / -1 = 0.

Using trigonometric identities

We can also use the periodicity and symmetry properties of the tangent function:

• The tangent function has a period of 180°, meaning tan(θ + 180°) = tan(θ).
• tan(0°) = 0, so tan(180°) = tan(0° + 180°) = tan(0°) = 0.

This confirms our earlier result using a different approach.

Step-by-step calculation

1. Recall the definition of tangent: tan(θ) = sin(θ) / cos(θ).
2. For θ = 180°:
   • sin(180°) = 0
   • cos(180°) = -1
3. Calculate tan(180°) = 0 / -1 = 0.

This step-by-step approach clearly shows how to arrive at the result.

Verification

To ensure our answer is correct, let's consider the graph of the tangent function:

• The tangent function has vertical asymptotes at θ = 90° + k*180° (where k is an integer).
• At θ = 180°, the function crosses the x-axis, confirming tan(180°) = 0.

This graphical verification supports our algebraic result.