Question

exact value of the unit circle function tan 240

Answer 1

To find the exact value of \(\tan(240^\circ)\), we first need to determine which quadrant \(240^\circ\) is in and then find the reference angle.

1. **Determine Quadrant**:
- \(240^\circ\) is in the third quadrant.

2. **Find Reference Angle**:
- The reference angle for \(240^\circ\) can be calculated as:
\[
240^\circ - 180^\circ = 60^\circ
\]

3. **Using the Unit Circle**:
- In the third quadrant, both sine and cosine are negative. The tangent function, which is the ratio of sine to cosine, will be positive because it is negative divided by negative.
- Therefore:
\[
\tan(240^\circ) = \tan(60^\circ)
\]
- We know that:
\[
\tan(60^\circ) = \sqrt{3}
\]
- Thus:
\[
\tan(240^\circ) = \tan(60^\circ) = \sqrt{3}
\]

Final Result:
\[
\boxed{\sqrt{3}}
\]