In basic trigonometry, what is the exact value of tan 150° expressed in standard surd form?

Introduction / Context:
This question tests your understanding of trigonometric values for special angles and how to use reference angles and quadrant signs. The angle 150° is not one of the primary 30° or 45° angles, but it is closely related to 30° through the identity tan(180° − θ) = −tan θ. Recognising this relationship makes the evaluation of tan 150° straightforward without a calculator.

Given Data / Assumptions:

• The angle of interest is 150°.
• We work in degrees, not radians.
• We use exact trigonometric values for special angles such as 30°, 45°, and 60°.
• The final answer should be written in standard surd form involving √3 where appropriate.

Concept / Approach:
The key idea is to express 150° in a form that uses a known reference angle. We note that 150° = 180° − 30°. For the tangent function, there is a useful identity: tan(180° − θ) = −tan θ. This tells us that tan 150° is simply the negative of tan 30°. Since tan 30° has a standard exact value 1/√3, we can immediately determine tan 150° by applying this identity and paying attention to the sign of tangent in the second quadrant, where 150° lies.

Step-by-Step Solution:
Write 150° as 180° − 30° to use the identity tan(180° − θ) = −tan θ.Apply the identity: tan 150° = tan(180° − 30°) = −tan 30°.Recall the standard value tan 30° = 1/√3.Substitute this value into the expression: tan 150° = −(1/√3).So the exact value of tan 150° in standard surd form is −1/√3.

Verification / Alternative check:
From the unit circle perspective, 150° lies in the second quadrant, where sine is positive and cosine is negative. Tangent is defined as sin θ / cos θ, which becomes positive/negative = negative in this quadrant. Since the reference angle is 30°, the magnitude of tan 150° must match tan 30°, but the sign must be negative. Therefore the answer must be −1/√3, confirming our earlier result using the formula tan(180° − θ) = −tan θ.

Why Other Options Are Wrong:

• −√3 is the tangent of angles like 120° or 300°, not 150°.
• 1/√3 is tan 30°, which is positive and would apply in the first quadrant, not at 150° in the second quadrant.
• √3 is the tangent of 60° and also has the wrong magnitude and sign for this angle.
• 0 is the tangent of 0° or 180°, not 150°.

Common Pitfalls:

• Forgetting that 150° is in the second quadrant where tangent is negative.
• Confusing the identity tan(180° − θ) = −tan θ with the formula for sine or cosine, which behave differently with supplementary angles.
• Mixing up the standard values of tan 30° and tan 60° and thus writing ±√3 instead of ±1/√3.

Final Answer:
-1/√3