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 81

Capitolo 4, Problema 81

Find each exact function value. See Example 3.
sin (-8π/ 3)

Guida verificata passo dopo passo

Recognize that the sine function is periodic with period \(2\pi\), meaning \(\sin(\theta) = \sin(\theta + 2k\pi)\) for any integer \(k\). This allows us to find an equivalent angle within the standard interval \([0, 2\pi)\) or \([-\pi, \pi)\) for easier evaluation.

Start with the given angle \(-\frac{8\pi}{3}\). To find a coterminal angle between \(0\) and \(2\pi\), add multiples of \(2\pi\) until the angle lies within this range. Specifically, add \(2\pi\) (which is \(\frac{6\pi}{3}\)) to \(-\frac{8\pi}{3}\):

\[-\frac{8\pi}{3} + 2\pi = -\frac{8\pi}{3} + \frac{6\pi}{3} = -\frac{2\pi}{3}.\]

Since \(-\frac{2\pi}{3}\) is still negative, add another \(2\pi\) to bring it into the positive range:

\[-\frac{2\pi}{3} + 2\pi = -\frac{2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3}.\]

Now, evaluate \(\sin\left(\frac{4\pi}{3}\right)\). Recall that \(\frac{4\pi}{3}\) is in the third quadrant where sine is negative, and its reference angle is \(\pi - \frac{4\pi}{3} = \frac{\pi}{3}\). Use the known sine value for \(\frac{\pi}{3}\) to find the exact value.

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Unit Circle and Angle Measurement in Radians

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Angles are often measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles on the unit circle, including negative and large angles, is essential for evaluating trigonometric functions like sine.

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Percorso guidato

06:11

Introduction to the Unit Circle

Angle Coterminality and Reduction

Angles that differ by full rotations (multiples of 2π) share the same terminal side and thus have the same trigonometric values. To find the exact value of functions like sin(-8π/3), reduce the angle by adding or subtracting 2π until it lies within the standard interval [0, 2π). This simplification helps in identifying the reference angle and corresponding function value.

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Percorso guidato

04:46

Coterminal Angles

Sine Function Properties and Exact Values

The sine function gives the y-coordinate of a point on the unit circle corresponding to an angle. It is periodic with period 2π and odd, meaning sin(-θ) = -sin(θ). Knowing exact sine values for common angles (like π/3, π/6, π/4) allows precise evaluation without a calculator, which is crucial for finding exact values of sine at given angles.

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Percorso guidato

5:53

Graph of Sine and Cosine Function

Pratica correlata

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For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.

s = 51

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Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.

s = ―7.4

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Find each exact function value. See Example 3.

cos 3π

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Find the exact values of s in the given interval that satisfy the given condition.

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