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

Verified step by step guidance

Understand that an angle in standard position is drawn with its vertex at the origin and its initial side along the positive x-axis.

Identify the given angle: \(\frac{8\pi}{3}\). Notice that this angle is greater than \(2\pi\), which means it makes more than one full rotation.

To find the equivalent angle between \(0\) and \(2\pi\), subtract multiples of \(2\pi\) from \(\frac{8\pi}{3}\) until the result lies within one full rotation: calculate \(\frac{8\pi}{3} - 2\pi\).

Simplify the subtraction: express \(2\pi\) as \(\frac{6\pi}{3}\) to get \(\frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3}\).

Draw the angle \(\frac{2\pi}{3}\) in standard position by starting from the positive x-axis and rotating counterclockwise by \(\frac{2\pi}{3}\) radians.

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Key Concepts

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

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The angle is measured by rotating the initial side to the terminal side, either counterclockwise for positive angles or clockwise for negative angles.

Radian Measure

Radian measure defines angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Since 2π radians equal 360 degrees, angles can be converted between radians and degrees for easier interpretation and drawing.

Coterminal Angles

Coterminal angles share the same terminal side but differ by full rotations of 2π radians. To draw an angle like 8π/3, it is often helpful to subtract multiples of 2π to find a coterminal angle between 0 and 2π, simplifying the drawing process.

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Coterminal Angles

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 8–13, 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.

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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, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋

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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.

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sec 45°

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