Tan 150 Without Calculator

Calculating tan(150°) without a calculator requires understanding trigonometric identities and reference angles. This guide explains the step-by-step method, provides a calculator, and helps you interpret the result.

How to calculate tan(150°)

To find tan(150°) without a calculator, follow these steps:

Identify the reference angle for 150°.
Determine the quadrant where 150° is located.
Use the tangent of the reference angle.
Apply the sign rule based on the quadrant.

The reference angle for 150° is 30° (180° - 150°). Since 150° is in the second quadrant, the tangent function is negative there. Therefore, tan(150°) = -tan(30°).

Formula used

tan(θ) = sin(θ)/cos(θ)

For angles in the second quadrant:

tan(180° - θ) = -tan(θ)

We use these identities to find tan(150°) by first calculating tan(30°) and then applying the sign rule.

Worked example

Let's calculate tan(150°):

Reference angle: 150° - 180° = 30°
Quadrant: Second quadrant (180°-270°)
tan(30°) = √3/3 ≈ 0.577
tan(150°) = -tan(30°) ≈ -0.577

The exact value is -√3/3, and the approximate value is -0.577.

Interpreting the result

The negative value indicates the angle is in the second quadrant where tangent is negative. The magnitude (0.577) represents the ratio of the opposite side to the adjacent side in a right triangle with this angle.

Remember that tan(150°) is not the same as tan(150). The latter represents a ratio of lengths, while the former is an angle in degrees.

FAQ

Why is tan(150°) negative?

Because 150° is in the second quadrant where tangent is negative. The reference angle is 30°, and we apply the sign rule for the quadrant.

What is the exact value of tan(150°)?

The exact value is -√3/3. This comes from tan(30°) = √3/3 and applying the sign rule for the second quadrant.

How does this relate to the unit circle?

On the unit circle, tan(150°) corresponds to the y-coordinate divided by the x-coordinate at that angle. The point is (-√3/2, 1/2).