Final answer:
The cosecant of an angle A given that θ = 17π/12 is found by first converting the angle to a standard angle, calculating the sine, and then finding its reciprocal. The correct answer, considering the provided options, is -2/&sqrt;2, matching option d) -2/&sqrt;2.
Step-by-step explanation:
The question involves finding the cosecant (csc) of an angle A given the angle θ = 17π/12 and the side a = 7. To find csc A, we first need to determine the value of sine (sin) for the angle because csc A is the reciprocal of sin A. Since the problem provides an angle in radians, we can find its sine using trigonometric functions.
However, the angle θ = 17π/12 is not a standard position angle, so we need to convert it into one of the standard angles that we know the sine for. Converting 17π/12 into a comparable standard angle, we find that it is equivalent to -π/12 (since 17π/12 is the same as -π/12 mod 2π). Looking at the unit circle, the sine of -π/12, which is the same as the sine of 11π/12, is -&sqrt;2/2.
Therefore, the csc A, which is the reciprocal of sin A, will be -2/&sqrt;2, which simplifies to -&sqrt;2. Among the given options, the correct answer is d) -2/&sqrt;2.
To find csc A, we need to know the value of A. In this case, θ = 17π/12 represents the angle A. The function csc is the reciprocal of sin, so csc A = 1/sin A. To find sin A, we can use the unit circle or a calculator. Since sin A can be positive or negative based on the quadrant of the angle, we need to consider the signs of the functions in each quadrant.
A = 17π/12 is in the second quadrant, where sin A is positive. We can find sin A by looking at the reference angle in the first quadrant, which is π/12. Sin π/12 = √2/4.
Since csc A = 1/sin A, csc A = 1/(√2/4) = 4/√2 = 2/√2 = -√2/2.
Therefore, the correct answer is a) -√2/3.