Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problema 15

Capitolo 4, Problema 15

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.
s = ―π

Guida verificata passo dopo passo

Identify the given angle: here, the angle is \(s = -\pi\) radians. This means the angle is negative, indicating a clockwise rotation of \(\pi\) radians from the positive x-axis.

Recall the unit circle values for \(\pi\) radians: on the unit circle, an angle of \(\pi\) radians corresponds to the point \((-1, 0)\), where the x-coordinate is \(\cos \pi\) and the y-coordinate is \(\sin \pi\).

Use the even-odd properties of sine and cosine to find values for negative angles: since \(\sin(-\theta) = -\sin \theta\) and \(\cos(-\theta) = \cos \theta\), apply these to \(s = -\pi\).

Calculate \(\sin s\) using the property: \(\sin(-\pi) = -\sin \pi\). Since \(\sin \pi = 0\), this gives \(\sin(-\pi) = 0\).

Calculate \(\cos s\) and \(\tan s\): \(\cos(-\pi) = \cos \pi = -1\), and \(\tan s = \frac{\sin s}{\cos s} = \frac{0}{-1} = 0\).

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Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles measured in radians correspond to points on the unit circle, where the x-coordinate gives the cosine value and the y-coordinate gives the sine value of the angle. Understanding how to locate angles like -π on the unit circle is essential for finding exact trigonometric values.

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Introduction to the Unit Circle

Sine, Cosine, and Tangent Functions

Sine and cosine functions represent the y and x coordinates, respectively, of a point on the unit circle corresponding to an angle. Tangent is the ratio of sine to cosine (tan s = sin s / cos s). Knowing these definitions helps in calculating exact values for any real number angle, including negative angles like -π.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Properties of Trigonometric Functions for Negative Angles

Trigonometric functions have specific symmetry properties: sine is an odd function (sin(-s) = -sin s), cosine is even (cos(-s) = cos s), and tangent is odd (tan(-s) = -tan s). These properties allow simplification when evaluating trigonometric values at negative angles such as -π.

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