Trigonometry: Question 6

Worked solution

Step 1: Locate the quadrant

$210°$ lies between $180°$ and $270°$, so it is in the third quadrant. In the third quadrant, sine is negative (only tangent is positive there).

Step 2: Find the reference angle

The reference angle is the acute angle between the terminal arm and the horizontal axis: $210° - 180° = 30°$

Step 3: Apply the exact value at the reference angle

The standard exact value is: $\sin 30° = \frac12$

Step 4: Apply the sign for the third quadrant

Since sine is negative in the third quadrant: $\sin 210° = -\sin 30° = -\frac12$

Why the other options are wrong

• B ($\tfrac12$): correct magnitude from the reference angle, but the negative sign for the third quadrant has been dropped.
• C ($-\tfrac{\sqrt3}{2}$): this is $-\cos 30°$, mixing up sine and cosine at the reference angle.
• D ($\tfrac{\sqrt3}{2}$): this is $\cos 30°$ with no sign applied at all. Both the ratio and the sign are wrong.

Final answer

• $\sin 210° = \boxed{-\dfrac12}$, option A.