In Exercises 8–13, find the exact value of each expression. Do not use a calculator. cot (-8𝜋/3)

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Recognize that the cotangent function is periodic with period \(\pi\), so we can simplify the angle \(-\frac{8\pi}{3}\) by adding or subtracting multiples of \(\pi\) to find a coterminal angle within a standard interval, such as \([0, 2\pi)\).

Add \(3\pi\) (which is \(\pi\) times 3) to \(-\frac{8\pi}{3}\) to find a positive coterminal angle: \(-\frac{8\pi}{3} + 3\pi = -\frac{8\pi}{3} + \frac{9\pi}{3} = \frac{\pi}{3}\).

Now, evaluate \(\cot\left(\frac{\pi}{3}\right)\). Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).

Use the known exact values for sine and cosine at \(\frac{\pi}{3}\): \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

Substitute these values into the cotangent formula: \(\cot\left(\frac{\pi}{3}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\), and simplify the fraction to find the exact value.

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주요 개념

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Cotangent Function

Cotangent is a trigonometric function defined as the ratio of the cosine to the sine of an angle, cot(θ) = cos(θ)/sin(θ). It is the reciprocal of the tangent function and is periodic with period π. Understanding cotangent helps in evaluating expressions involving cot(θ) without a calculator.

Angle Reduction Using Coterminal Angles

Angles differing by full rotations (multiples of 2π) are coterminal and have the same trigonometric values. To simplify cot(-8π/3), add or subtract multiples of 2π to find an equivalent angle within the standard interval [0, 2π) or (-π, π]. This step is crucial for evaluating trigonometric functions exactly.

Reference Angles and Sign Determination

After reducing the angle to a standard position, identify its reference angle in the first quadrant to find exact trigonometric values. Also, determine the sign of the function based on the quadrant where the angle lies, using the ASTC (All Students Take Calculus) rule. This ensures the correct exact value of cotangent.