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

Verified step by step guidance

1. Start by understanding the expression \( \cot(\csc^{-1}(8)) \). This involves finding the cotangent of an angle whose cosecant is 8.
2. Let \( \theta = \csc^{-1}(8) \). This means that \( \csc(\theta) = 8 \). Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \), so \( \sin(\theta) = \frac{1}{8} \).
3. Use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find \( \cos(\theta) \). Substitute \( \sin(\theta) = \frac{1}{8} \) into the identity to get \( \left(\frac{1}{8}\right)^2 + \cos^2(\theta) = 1 \).
4. Solve for \( \cos(\theta) \) by simplifying the equation: \( \frac{1}{64} + \cos^2(\theta) = 1 \). This gives \( \cos^2(\theta) = 1 - \frac{1}{64} \).
5. Finally, find \( \cot(\theta) \) using \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). Substitute the values of \( \cos(\theta) \) and \( \sin(\theta) \) to find the exact value of \( \cot(\theta) \).

Key Concepts

Cosecant and Its Inverse

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.

Cotangent Function

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.

Right Triangle Relationships

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).