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

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Recognize that the expression is \( \cot(\csc^{-1} 8) \). Let \( \theta = \csc^{-1} 8 \), which means \( \csc \theta = 8 \).

Recall that \( \csc \theta = \frac{1}{\sin \theta} \), so \( \sin \theta = \frac{1}{8} \).

Draw a right triangle where the angle \( \theta \) has an opposite side of length 1 and a hypotenuse of length 8, based on \( \sin \theta = \frac{1}{8} \).

Use the Pythagorean theorem to find the adjacent side: \( \text{adjacent} = \sqrt{8^2 - 1^2} = \sqrt{64 - 1} = \sqrt{63} \).

Calculate \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{\sqrt{63}}{1} = \sqrt{63} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosecant Function (csc⁻¹)

The inverse cosecant function, csc⁻¹(x), returns the angle whose cosecant is x. Since cosecant is the reciprocal of sine, csc⁻¹(8) gives an angle θ such that sin(θ) = 1/8. Understanding this helps in converting the inverse trigonometric expression into a more manageable angle.

Relationship Between Trigonometric Ratios

Cotangent is the reciprocal of tangent and can be expressed as cos(θ)/sin(θ). Knowing how cotangent relates to sine and cosine allows you to find cot(csc⁻¹(8)) by first determining sin(θ) and cos(θ) from the given angle θ.

Right Triangle Sketch and Pythagorean Theorem

Sketching a right triangle based on the given trigonometric ratio helps visualize the problem. Using sin(θ) = opposite/hypotenuse = 1/8, you can assign side lengths and apply the Pythagorean theorem to find the adjacent side, enabling calculation of cotangent as adjacent/opposite.

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