evaluate csc 7pi/6

1/sin (7pi/6)

7 pi/6 is pi/6 or 30 degrees below -y axis in quadrant 3
so
sin 7 pi/6 = -1/2

inverse is csc = -2

To evaluate the csc (cosecant) of an angle, we need to find the reciprocal of the sine of that angle.

The angle given is 7π/6.

To find the sine of 7π/6, we need to determine the reference angle. The reference angle is the angle formed between the terminal side of the given angle (7π/6) and the x-axis.

In this case, 7π/6 is in the third quadrant, where the sine is negative.

The reference angle can be found by subtracting the given angle from the nearest multiple of π. In this case, the nearest multiple of π is 2π, so the reference angle is 2π - (7π/6) = (12π/6) - (7π/6) = 5π/6.

Now we can find the sine of the reference angle, which is sin(5π/6) = √3/2.

Since the sine is negative in the third quadrant, the sine of 7π/6 is -√3/2.

Finally, we can find the cosecant by taking the reciprocal of the sine: csc(7π/6) = 1 / sin(7π/6) = 1 / (-√3/2) = -2/√3.

Therefore, the value of csc(7π/6) is -2/√3.

To evaluate the cosecant (csc) of an angle, we need to use the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.

To find the value of csc(7π/6), we first need to identify the angle on the unit circle.

In this case, the angle is 7π/6 radians, approximately equal to 210 degrees (since π radians is equivalent to 180 degrees).

To understand the value of csc, we can recall that csc is the reciprocal of the sine function. The sine function represents the y-coordinate on the unit circle.

Now, let's proceed to find csc(7π/6):

1. Draw a unit circle and mark the angle of 7π/6 radian or 210 degrees.
2. Notice that the reference angle of 7π/6 radian, which is a positive angle between the terminal side and the x-axis, is π/6 radian or 30 degrees. It can be found by subtracting the given angle from 2π or 360 degrees.
3. The sine function value of π/6 is 1/2.
4. As csc is the reciprocal of sin, the csc value of π/6 is the reciprocal of 1/2, which is 2.

Therefore, csc(7π/6) equals 2.