Evaluate cot(11pi/6) to the nearest hundredth (given in radians).
in the decimal form they asked for:
-1.73
11pi/6 is in the fourth quadrant, in the fourth quadrant the tangent is negative, so the cotangent is negative.
The angle is pi/6 away from the x-axis, so the "angle in standard position" is pi/6
tan pi/6 = 1/�ã3, so cot pi/6 = �ã3
then cot(11pi/6) = -�ã3
Remark
pi/6 rad = 30�‹
tan pi/6 = 1/sqrt(3), so cot pi/6 = sqrt(3)
then cot(11pi/6) = -sqrt(3)
Remark
pi/6 rad = 30 deg
To evaluate cot(11π/6), we need to know the values of sine (sin) and cosine (cos) of 11π/6.
First, let's find the value of sin(11π/6):
Since 11π/6 is equivalent to 2π + π/6, we know that the angle lies in the second quadrant. In the second quadrant, the value of sin is positive.
We can determine the value of sin(π/6) using the special triangle for the 30-60-90 degree triangle. In this triangle, the opposite side's length is 1, and the hypotenuse's length is 2. Therefore, sin(π/6) = opposite/hypotenuse = 1/2.
Since the angle lies in the second quadrant, the value of sin(11π/6) is positive, so sin(11π/6) = sin(π/6) = 1/2.
Next, let's find the value of cos(11π/6):
Using the same triangle as before, we can determine the value of cos(π/6) = adjacent/hypotenuse = √3/2.
Since 11π/6 is equivalent to 2π + π/6, the angle lies in the second quadrant. In the second quadrant, the value of cos is negative.
Therefore, cos(11π/6) = -cos(π/6) = -√3/2.
Finally, to find cot(11π/6), we use the identity cot(x) = cos(x)/sin(x):
cot(11π/6) = -cos(11π/6)/sin(11π/6)
Substituting the previously determined values:
cot(11π/6) = -(-√3/2) / (1/2)
Simplifying:
cot(11π/6) = √3
Therefore, cot(11π/6) ≈ 1.73 (to the nearest hundredth).