Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 12

Chapter 1, Problem 12

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

Verified step by step guidance

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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Key Concepts

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

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.

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Related Practice

Textbook Question

In Exercises 8–13, find the exact value of each expression. Do not use a calculator. sec 22𝜋 3

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Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.

6 3 2 3 6 6 3 2 3 6

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

<IMAGE>

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.

csc 7𝜋/6

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Textbook Question

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. csc 𝜋

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Textbook Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

<IMAGE>

csc 45°

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Textbook Question

In Exercises 8–12, draw each angle in standard position. -135°

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Textbook Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

<IMAGE>

sec 45°

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