In Exercises 63–82, use a sketch to find the exact value of each expression.cot (csc⁻¹ 8)

The cosecant function, denoted as csc, is the reciprocal of the sine function. The inverse cosecant function, csc⁻¹, gives the angle whose cosecant is a given value. In this case, csc⁻¹(8) represents the angle θ such that csc(θ) = 8, which implies sin(θ) = 1/8.

The cotangent function, denoted as cot, is the reciprocal of the tangent function. It can be expressed as cot(θ) = cos(θ)/sin(θ). To find cot(csc⁻¹(8)), we need to determine the cosine and sine values of the angle θ derived from csc⁻¹(8) and then compute their ratio.

In trigonometry, right triangles are fundamental for understanding the relationships between angles and side lengths. Given csc(θ) = 8, we can visualize a right triangle where the hypotenuse is 8 and the opposite side is 1 (since sin(θ) = 1/8). Using the Pythagorean theorem, we can find the adjacent side, which is essential for calculating cot(θ).