To find the exact value of csc 11pi/6, first determine sin 11pi/6. The angle 11pi/6 is in Quadrant IV, where sine is negative. The reference angle is pi/6, and sin pi/6 equals 1/2. Therefore, sin 11pi/6 is -1/2. Since csc x is 1/sin x, csc 11pi/6 is 1/(-1/2), which simplifies to -2.
The reciprocal of sin(11pi/6) is defined as the cosecant of 11pi/6. To find this value, first determine sin(11pi/6). Since 11pi/6 is in the fourth quadrant, sin(11pi/6) equals -sin(pi/6), which is -1/2. Therefore, csc(11pi/6) is the reciprocal of -1/2, resulting in an exact value of -2.
To find csc(-pi/6), recall cosecant is an odd function, meaning csc(-x) = -csc(x). First, determine sin(-pi/6). As -pi/6 is coterminal with 11pi/6, or simply -sin(pi/6), its sine is -1/2. Thus, csc(-pi/6) is the reciprocal of -1/2, which results in the value of -2.
The angle 11pi/6 radians is located in the fourth quadrant of the unit circle. To find its sine, we can use its reference angle, which is pi/6. In the fourth quadrant, the sine function is negative. Therefore, sin(11pi/6) is equal to -sin(pi/6). Knowing that sin(pi/6) is 1/2, the exact value of sin(11pi/6) is -1/2.
Cosecant and sine are reciprocal trigonometric functions. Specifically, csc(x) is defined as 1 divided by sin(x), provided that sin(x) is not zero. Therefore, csc(11pi/6) is the reciprocal of sin(11pi/6). If sin(11pi/6) is -1/2, then csc(11pi/6) would be 1 divided by -1/2, resulting in -2.
The angle 11pi/6 radians lies in the fourth quadrant, as it's equivalent to 330 degrees, between 270 and 360 degrees. In the fourth quadrant, the sine function is negative. Since cosecant is the reciprocal of sine, csc(11pi/6) will also be negative. Knowing sin(11pi/6) is -1/2, csc(11pi/6) equals -2.
On the unit circle, the angle 11pi/6 corresponds to a point with coordinates (sqrt(3)/2, -1/2). The y-coordinate represents the sine of the angle. Therefore, sin(11pi/6) is -1/2. Since cosecant is the reciprocal of sine, csc(11pi/6) is 1 divided by -1/2. This calculation yields an exact value of -2.
To find csc(330 degrees), recognize that 330 degrees is equivalent to 11pi/6 radians. This angle is in the fourth quadrant, where sine is negative. The reference angle is 30 degrees. Thus, sin(330 degrees) is -sin(30 degrees), which is -1/2. Since cosecant is the reciprocal of sine, csc(330 degrees) equals 1/(-1/2), resulting in -2.
No, csc(x) cannot be strictly between -1 and 1 (excluding 0). Cosecant is the reciprocal of sine. The range of sin(x) is [-1, 1]. Any non-zero sine value within this range, when reciprocated, will result in a value greater than or equal to 1, or less than or equal to -1. For example, csc(11pi/6) is...
The reference angle for 11pi/6 is pi/6 (30 degrees), as 11pi/6 is 2pi minus pi/6. This angle lies in the fourth quadrant. To find csc(11pi/6), we first determine sin(pi/6), which is 1/2. Since sine is negative in Quadrant IV, sin(11pi/6) becomes -1/2. Therefore, csc(11pi/6), being its reciprocal, is -2.
The value of csc(11pi/6 + 2pi) is identical to csc(11pi/6) because the cosecant function has a period of 2pi. Adding or subtracting multiples of 2pi to an angle results in a coterminal angle, which has the same trigonometric values. As established, csc(11pi/6) is -2. Therefore, csc(11pi/6 + 2pi) also equals -2.