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 19

Chapter 1, Problem 19

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.cos (-𝜋/6)

Verified step by step guidance

Identify the angle \(-\pi/6\) on the unit circle. The corresponding positive angle is \(11\pi/6\).

Locate the point on the unit circle for \(11\pi/6\), which is \((\sqrt{3}/2, -1/2)\).

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. Therefore, \(\cos(-\pi/6) = \sqrt{3}/2\).

Use the even property of cosine: \(\cos(-\theta) = \cos(\theta)\).

Thus, \(\cos(-\pi/6) = \cos(\pi/6)\), confirming the value found in step 3.

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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.

Even and Odd Properties of Trigonometric Functions

Trigonometric functions exhibit specific symmetry properties: cosine is an even function, meaning cos(-x) = cos(x), while sine is an odd function, meaning sin(-x) = -sin(x). These properties allow for simplifications when evaluating trigonometric functions at negative angles, making it easier to find values using known angles on the unit circle.

Trigonometric Function Values

The values of trigonometric functions such as sine and cosine can be derived from the coordinates of points on the unit circle. For example, for an angle of -π/6, the corresponding point on the unit circle provides the cosine and sine values directly. Understanding how to extract these values from the unit circle is crucial for solving trigonometric problems efficiently.

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Related Practice

Textbook Question

In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators.sin θ = 3/5, cos θ = 4/5

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Textbook Question

In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators.__sin θ = 6, cos θ = √137 7

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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 𝜋/4 + csc 𝜋/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.tan 3𝜋/2

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Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.cos 𝜋/3