Tan 120 Without Calculator

Calculating tan(120) without a calculator requires understanding of trigonometric identities and reference angles. This guide explains step-by-step methods to find the tangent of 120 degrees, including using the unit circle and trigonometric identities.

How to Calculate tan(120)

The tangent of 120 degrees can be found using several methods. The most common approaches are using trigonometric identities and the reference angle method. Both methods rely on the fact that 120 degrees is in the second quadrant of the unit circle.

tan(θ) is defined as the ratio of the y-coordinate to the x-coordinate of a point on the unit circle at angle θ.

Since 120° is in the second quadrant, we know that:

Sine is positive
Cosine is negative
Tangent is negative (since tan = sin/cos)

Using Trigonometric Identities

One way to find tan(120) is by using the tangent addition formula:

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

We can express 120° as 180° - 60° and use the identity for tan(180° - θ):

tan(180° - θ) = -tanθ

Therefore:

tan(120°) = tan(180° - 60°) = -tan(60°)

We know that tan(60°) = √3, so:

tan(120°) = -√3 ≈ -1.73205

Reference Angle Method

The reference angle method involves finding the reference angle for 120° and using the properties of the unit circle.

For angles in the second quadrant (90° < θ < 180°), the reference angle (θ') is calculated as:

θ' = 180° - θ

For 120°:

θ' = 180° - 120° = 60°

We know the coordinates for 60° on the unit circle are (cos60°, sin60°) = (0.5, √3/2). In the second quadrant, the x-coordinate is negative while the y-coordinate remains positive:

(x, y) = (-0.5, √3/2)

Therefore, tan(120°) is the ratio of y to x:

tan(120°) = (√3/2) / (-0.5) = -√3 ≈ -1.73205

Worked Example

Let's calculate tan(120°) using both methods to verify our result.

Method 1: Using Trigonometric Identities

Express 120° as 180° - 60°
Use the identity tan(180° - θ) = -tanθ
Calculate tan(60°) = √3
Therefore, tan(120°) = -√3 ≈ -1.73205

Method 2: Reference Angle Method

Find the reference angle: 180° - 120° = 60°
Find the coordinates for 60°: (0.5, √3/2)
Adjust for the second quadrant: (-0.5, √3/2)
Calculate tan(120°) = (√3/2) / (-0.5) = -√3 ≈ -1.73205

Both methods yield the same result, confirming that tan(120°) = -√3.

Common Mistakes

When calculating tan(120°) without a calculator, it's easy to make the following mistakes:

Forgetting the sign: Since 120° is in the second quadrant where tangent is negative, forgetting the negative sign will give an incorrect result.
Using the wrong reference angle: Calculating the reference angle incorrectly (e.g., using 120° instead of 60°) will lead to wrong coordinates.
Incorrectly applying identities: Misapplying trigonometric identities like tan(180° - θ) = -tanθ can result in errors.

Always double-check the quadrant of the angle and the sign of trigonometric functions in that quadrant.

FAQ

Is tan(120°) positive or negative?: tan(120°) is negative because 120° is in the second quadrant where tangent is negative.
Can I use a calculator to verify my result?: Yes, you can use a calculator to verify that tan(120°) ≈ -1.73205, which matches our manual calculation.
What is the exact value of tan(120°)?: The exact value is -√3, which is approximately -1.73205.
How do I find tan(120°) using a calculator?: Enter 120° in the tangent function of your calculator to get the result directly.
What are the coordinates on the unit circle for 120°?: The coordinates are (-0.5, √3/2), which can be used to calculate tan(120°).