tan of 2pi/3
Understand the Problem
The question is asking for the tangent value of the angle 2π/3 radians. To solve this, we can recognize that 2π/3 radians corresponds to an angle in the second quadrant where tangent values are negative.
Answer
$$ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} $$
Answer for screen readers
The final answer is
$$ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} $$
Steps to Solve
- Identify the angle in degrees
Convert the angle from radians to degrees to better understand its position. The conversion formula is:
$$ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} $$
For ( \frac{2\pi}{3} ) radians:
$$ \text{Degrees} = \frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ $$
- Determine the reference angle
The reference angle is the angle formed with the x-axis. For ( 120^\circ ):
$$ \text{Reference angle} = 180^\circ - 120^\circ = 60^\circ $$
- Find the tangent of the reference angle
The tangent of the reference angle ( 60^\circ ) is:
$$ \tan(60^\circ) = \sqrt{3} $$
- Determine the sign of the tangent
Since ( 120^\circ ) is in the second quadrant where tangent values are negative, we have:
$$ \tan(120^\circ) = -\tan(60^\circ) = -\sqrt{3} $$
- Final result
Thus, the tangent of ( \frac{2\pi}{3} ) radians is:
$$ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} $$
The final answer is
$$ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} $$
More Information
The tangent function relates to the ratio of the opposite side to the adjacent side in a right triangle. When dealing with angles in different quadrants, it’s key to remember the sign changes based on the quadrant's characteristics.
Tips
- Confusing the sign of the tangent when identifying the quadrant: Remember that tangent is negative in the second quadrant.
