Tan(5π/3)
To find the value of tan(5π/3), we will use the unit circle and trigonometric ratios
To find the value of tan(5π/3), we will use the unit circle and trigonometric ratios.
First, let’s understand the angle (5π/3) in the coordinate plane.
1 full rotation in radians is equal to 2π radians. So, 5π/3 represents an angle that is more than one full rotation, but less than two full rotations.
To determine the exact value of the angle, we can use the fact that 5π/3 is equivalent to (4π/3 + π/3).
Now, let’s look at the coordinates of the angles (4π/3) and (π/3) on the unit circle:
Angle (4π/3):
– The point on the unit circle for (4π/3) is (-1/2, -√3/2) (in the third quadrant), meaning the x-coordinate is -1/2 and the y-coordinate is -√3/2.
Angle (π/3):
– The point on the unit circle for (π/3) is (1/2, √3/2) (in the first quadrant), meaning the x-coordinate is 1/2 and the y-coordinate is √3/2.
Since tan is defined as the ratio of the sine to the cosine, we can calculate the tangent of (5π/3) as follows:
tan(5π/3) = [tan(4π/3 + π/3)]
= [(tan(4π/3) + tan(π/3))] / [1 – tan(4π/3)tan(π/3)]
= [(-√3/3 + √3/3)] / [1 – (-√3/3)(√3/3)]
= [0] / [1 – (-1/3)]
= 0 / (4/3)
= 0.
Therefore, the value of tan(5π/3) is 0.
Note: The value of tan(5π/3) can also be found directly by noticing that the angle (5π/3) is equivalent to an angle of 0 radians (or 0°). In the unit circle, the tangent function has a value of 0 at these angles, which is why tan(5π/3) is 0.
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