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 14

Chapter 1, Problem 14

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

Verified step by step guidance

Identify the angle \( \frac{5\pi}{3} \) on the unit circle.

Locate the corresponding point on the unit circle for \( \frac{5\pi}{3} \), which is \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \).

Recall that the secant function, \( \sec(\theta) \), is the reciprocal of the cosine function, \( \cos(\theta) \).

Determine the cosine of \( \frac{5\pi}{3} \) using the x-coordinate of the point, which is \( \frac{1}{2} \).

Calculate \( \sec(\frac{5\pi}{3}) = \frac{1}{\cos(\frac{5\pi}{3})} = \frac{1}{\frac{1}{2}} \).

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Key Concepts

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

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured in radians, allowing for easy calculation of trigonometric functions.

Trigonometric Functions

Trigonometric functions, including sine, cosine, tangent, secant, cosecant, and cotangent, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, these functions can be defined using the coordinates of points on the circle. For example, the secant function is the reciprocal of the cosine function, which can be derived from the x-coordinate of a point on the unit circle.

Radians and Angle Measurement

Radians are a unit of angular measure used in mathematics, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. The unit circle divides the circle into angles measured in radians, which are essential for defining trigonometric functions. Understanding how to convert between degrees and radians is crucial for solving problems involving trigonometric functions.

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Related Practice

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.

sin 𝜋/4 - cos 𝜋/4

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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.

sec 11𝜋/6

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Textbook Question

In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. -445°

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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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tan 𝜋/3

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Textbook Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

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>

sin 3𝜋/2

Textbook Question

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

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