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

Verified step by step guidance

Recall that the cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).

Identify the angle given: \(\pi\) radians, which is a quadrantal angle located on the negative x-axis of the unit circle.

Evaluate \(\sin \pi\). Since \(\sin \pi = 0\), substitute this value into the reciprocal expression for cosecant.

Since \(\csc \pi = \frac{1}{\sin \pi} = \frac{1}{0}\), recognize that division by zero is undefined.

Conclude that \(\csc \pi\) is undefined because the sine of \(\pi\) is zero, making the reciprocal undefined.

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Key Concepts

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

Quadrantal Angles

Quadrantal angles are angles that lie on the x- or y-axis in the unit circle, typically multiples of π/2 (e.g., 0, π/2, π, 3π/2). These angles have special sine and cosine values, often 0, ±1, which affect the evaluation of trigonometric functions.

Cosecant Function (csc)

The cosecant function is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). It is undefined wherever sin(θ) = 0, which commonly occurs at quadrantal angles like 0, π, and 2π.

Evaluating Trigonometric Functions at Specific Angles

To evaluate trigonometric functions at specific angles, substitute the angle into the function and use known sine and cosine values from the unit circle. For quadrantal angles, check if the function is defined or undefined due to division by zero.

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Related Practice

Textbook Question

In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 1 meter Arc Length, s: 600 centimeters

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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.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3

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.

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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

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

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