what is the exact value of tan- 13pi/6
13pi/6 = 2pi + pi/6
so, tan 13pi/6 = tan pi/6 = 1/√3
tan -13pi/6 = tan -pi/6 = -1/√3
To find the exact value of tan(-13π/6), we need to locate the angle in the unit circle and evaluate its tangent.
First, let's convert the angle -13π/6 into its equivalent positive angle. -13π/6 is equivalent to an angle of 11π/6, or 2π + (11π/6).
Next, we represent 11π/6 on the unit circle. To do this, we divide the circle into six equal sections (since the denominator of the angle is 6) and count clockwise from the positive x-axis.
Starting from the positive x-axis, we move π/6 (or 30 degrees) in a counter-clockwise direction to reach the first section. We then continue moving 10 more sections in a clockwise direction to reach the angle of 11π/6.
The point where the angle 11π/6 intersects the unit circle corresponds to the coordinates (-√3/2, -1/2).
Now, we evaluate the tangent of the angle by dividing the y-coordinate by the x-coordinate. In this case, tan(11π/6) is equal to (-1/2) / (-√3/2).
To simplify this further, we multiply both the numerator and denominator by 2/√3, which gives us -1 / -√3. Simplifying again, we get √3/3.
Therefore, the exact value of tan(-13π/6) is √3/3.
To find the exact value of tan(-13π/6), we can use the trigonometric identity:
tan(-θ) = -tan(θ)
In this case, θ = 13π/6, so:
tan(-13π/6) = -tan(13π/6)
Now, let's determine the reference angle for 13π/6. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis.
To find the reference angle, we subtract the nearest full revolution (2π radians or 360 degrees) from the angle:
13π/6 - 2π = 13π/6 - 12π/6 = π/6
So, the reference angle for 13π/6 is π/6.
Since the terminal side of the angle lies in the fourth quadrant, the tangent value will be negative.
Now, we can find the tangent of the reference angle π/6:
tan(π/6) = sin(π/6) / cos(π/6)
The sine of π/6 is 1/2, and the cosine of π/6 is √3/2, so:
tan(π/6) = (1/2) / (√3/2) = 1/√3 = √3/3
Since the original angle was in the fourth quadrant, the tangent will be negative:
tan(-13π/6) = -√3/3
Therefore, the exact value of tan(-13π/6) is -√3/3.