Final Answer:
The exact value of the trigonometric function csc(7π/6) is √3/2.
Step-by-step explanation:
In trigonometry, the cosecant function, denoted as csc, is the reciprocal of the sine function. The formula for cosecant is csc(θ) = 1/sin(θ). To find csc(7π/6), we first need to determine the sine of 7π/6.
The angle 7π/6 is in the third quadrant of the unit circle, where the sine function is negative. The reference angle for 7π/6 is π/6, and sin(π/6) is 1/2. Since sine is negative in the third quadrant, sin(7π/6) = -1/2.
Now, we can use the reciprocal relationship to find csc(7π/6). csc(7π/6) = 1/sin(7π/6) = 1/(-1/2) = -2. However, we want the exact value, so we rationalize the denominator by multiplying the numerator and denominator by 2 to get -2 * 2/1 * 2 = -4/2. Simplifying this fraction further, we get -2.
Therefore, the exact value of csc(7π/6) is √3/2.
This result is obtained by recognizing the special trigonometric values associated with common angles on the unit circle. In this case, understanding the sine of π/6 and the reciprocal relationship allowed us to find the exact value without the need for a calculator.