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

Verified step by step guidance

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

Since the angle given is -135°, recognize that the negative sign indicates a clockwise rotation from the positive x-axis.

Measure 135° clockwise starting from the positive x-axis to locate the terminal side of the angle.

Note that rotating 135° clockwise is equivalent to rotating 225° counterclockwise (since 360° - 135° = 225°), so the terminal side lies in the third quadrant.

Draw the angle by first drawing the initial side along the positive x-axis, then rotating clockwise 135° to draw the terminal side, which will point diagonally in the third quadrant.

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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.

Measuring Negative Angles

Negative angles are measured by rotating the initial side clockwise from the positive x-axis. For example, a -135° angle means rotating 135 degrees clockwise, which places the terminal side in the third quadrant of the coordinate plane.

Quadrants and Angle Placement

The coordinate plane is divided into four quadrants, each corresponding to a range of angles. Understanding which quadrant an angle's terminal side lies in helps in accurately drawing and interpreting the angle in standard position.

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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 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.

sec 11𝜋/6

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Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.6 3 2 3 6 6 3 2 3 6Use 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.sec 5𝜋/3

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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>

tan 𝜋/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

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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