The value of `tan840^(@)` is equal to

Options

A. `(1)/(sqrt(3))`
B. `-(1)/(sqrt(3))`
C. `sqrt(3)`
D. `-sqrt(3)`

Answer

To find the value of \( \tan 840^\circ \), we can follow these steps:

### Step 1: Reduce the angle
First, we note that the tangent function has a periodicity of \( 180^\circ \). This means:
\[ \tan(θ) = \tan(θ + n \cdot 180^\circ) \]
for any integer \( n \). To reduce \( 840^\circ \), we can subtract \( 720^\circ \) (which is \( 4 \times 180^\circ \)):
\[ 840^\circ - 720^\circ = 120^\circ \]
Thus, we have:
\[ \tan(840^\circ) = \tan(120^\circ) \]

### Step 2: Find \( \tan 120^\circ \)
Next, we need to find \( \tan 120^\circ \). We know that:
\[ 120^\circ = 180^\circ - 60^\circ \]
Using the tangent subtraction identity:
\[ \tan(180^\circ - θ) = -\tan(θ) \]
we can write:
\[ \tan(120^\circ) = -\tan(60^\circ) \]

### Step 3: Calculate \( \tan 60^\circ \)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Therefore:
\[ \tan(120^\circ) = -\tan(60^\circ) = -\sqrt{3} \]

### Step 4: Final Result
Thus, we find:
\[ \tan(840^\circ) = \tan(120^\circ) = -\sqrt{3} \]

### Conclusion
The value of \( \tan 840^\circ \) is:
\[ -\sqrt{3} \approx -1.732 \]

---