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 10

Chapter 1, Problem 10

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

Verified step by step guidance

Recognize that the angle given is a quadrantal angle, specifically \(\pi\) radians, which corresponds to 180 degrees on the unit circle.

Recall that the tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Evaluate \(\sin \pi\) and \(\cos \pi\) using the unit circle values: \(\sin \pi = 0\) and \(\cos \pi = -1\).

Substitute these values into the tangent formula: \(\tan \pi = \frac{0}{-1}\).

Simplify the fraction to find the value of \(\tan \pi\), noting that division by a nonzero number is defined.

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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 coordinate plane, typically multiples of 90° or π/2 radians. These angles include 0, π/2, π, 3π/2, and 2π, where trigonometric functions often take special values or become undefined.

Tangent Function at Quadrantal Angles

The tangent function is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). At quadrantal angles, since cosine or sine can be zero, the tangent may be zero, a finite number, or undefined if division by zero occurs.

Evaluating Trigonometric Functions Using the Unit Circle

The unit circle provides coordinates (cos θ, sin θ) for any angle θ. Evaluating trigonometric functions at quadrantal angles involves identifying these coordinates and applying definitions, which helps determine exact values or identify undefined expressions.

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Related Practice

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.

tan 0

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Textbook Question

In Exercises 8–12, draw each angle in standard position. 8𝜋 3

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Textbook Question

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

cos 30°

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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.

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

tan 30°

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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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