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 23

Capitolo 4, Problema 23

Find each exact function value. See Example 2. cos (―4π/3)

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Recall that the cosine function is periodic with period \(2\pi\), so \(\cos(\theta) = \cos(\theta + 2k\pi)\) for any integer \(k\). This can help simplify the angle if needed.

Identify the angle \(-\frac{4\pi}{3}\) on the unit circle. Since the angle is negative, it means we rotate clockwise from the positive x-axis.

Convert the negative angle to a positive coterminal angle by adding \(2\pi\): \(-\frac{4\pi}{3} + 2\pi = \frac{2\pi}{3}\).

Evaluate \(\cos\left(\frac{2\pi}{3}\right)\) by recognizing that \(\frac{2\pi}{3}\) is in the second quadrant where cosine values are negative, and it corresponds to a reference angle of \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\).

Use the known cosine value for the reference angle \(\frac{\pi}{3}\), which is \(\frac{1}{2}\), and apply the sign based on the quadrant to find \(\cos\left(-\frac{4\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\).

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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 in trigonometry are often measured in radians, where 2π radians equal 360 degrees. Understanding how to locate angles like -4π/3 on the unit circle helps determine the corresponding coordinates and trigonometric values.

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Percorso guidato

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Introduction to the Unit Circle

Reference Angles and Quadrants

Reference angles are the acute angles formed between the terminal side of a given angle and the x-axis. Knowing the quadrant where the angle lies is essential because the signs of sine and cosine depend on the quadrant. For negative angles, rotation is clockwise, affecting the quadrant placement.

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Reference Angles on the Unit Circle

Cosine Function on the Unit Circle

The cosine of an angle corresponds to the x-coordinate of the point on the unit circle at that angle. To find cos(-4π/3), identify the point on the unit circle at -4π/3 radians and read its x-coordinate. This value gives the exact cosine function value.

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Sine, Cosine, & Tangent on the Unit Circle

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