Final answer:
The exact value of cosecant of the angle 23π/4 is the reciprocal of the sine of its equivalent angle 7π/4, which is √2/2. Thus, the cosecant is √2, or option C.
Step-by-step explanation:
The question asks for the exact value of the cosecant of the angle which is 23 times pi over 4. The angle can be simplified within the unit circle by finding an equivalent angle between 0 and 2π by dividing 23/4 by 2 and taking the remainder (since the cosecant function is periodic with a period of 2π in radians).
This gives us an equivalent angle of 7π/4. The cosecant function is the reciprocal of the sine function, so we need the sine of 7π/4, which is √2/2. The cosecant, being the reciprocal, would then be √2/(√2/2), which simplifies to 2/√2, and rationalizing the denominator gives √2. Therefore, the exact value of cosecant of 23π/4 is √2 (option C).