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 31

Capitolo 4, Problema 31

Find each exact function value. See Example 2.
tan 5π/6

Guida verificata passo dopo passo

Recall that the tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Identify the reference angle for \(\frac{5\pi}{6}\). Since \(\frac{5\pi}{6} = \pi - \frac{\pi}{6}\), the reference angle is \(\frac{\pi}{6}\).

Determine the signs of sine and cosine in the second quadrant (where \(\frac{5\pi}{6}\) lies). In the second quadrant, sine is positive and cosine is negative.

Use the known exact values for sine and cosine at the reference angle \(\frac{\pi}{6}\): \(\sin \frac{\pi}{6} = \frac{1}{2}\) and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).

Apply the signs and calculate \(\tan \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}} = \frac{\sin \frac{\pi}{6}}{-\cos \frac{\pi}{6}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\). Simplify this expression to find the exact value.

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Unit Circle and Reference Angles

The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Reference angles help find the function values by relating any angle to an acute angle in the first quadrant, simplifying calculations.

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Reference Angles on the Unit Circle

Tangent Function Definition

Tangent of an angle in the unit circle is the ratio of the y-coordinate to the x-coordinate (sin θ / cos θ). It can also be understood as the slope of the line formed by the angle, and its sign depends on the quadrant where the angle lies.

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Introduction to Tangent Graph

Evaluating Trigonometric Functions at Special Angles

Special angles like π/6, π/4, and π/3 have known exact sine, cosine, and tangent values. Using these known values and the angle’s quadrant, one can determine the exact value of trigonometric functions for angles like 5π/6.

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Evaluate Composite Functions - Special Cases

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