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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 49
Capitolo 3, Problema 49

Give the exact value of each expression. See Example 5. tan 30°

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1
Recall the definition of the tangent function in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Identify the values of \(\sin 30^\circ\) and \(\cos 30^\circ\) using known special angles: \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
Substitute these values into the tangent formula: \(\tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\).
Simplify the complex fraction by multiplying numerator and denominator appropriately: \(\tan 30^\circ = \frac{1}{2} \times \frac{2}{\sqrt{3}}\).
Further simplify the expression to get the exact value of \(\tan 30^\circ\) in simplest radical form.

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Definition of the Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. It can also be expressed as tan(θ) = sin(θ)/cos(θ), linking it to the sine and cosine functions.
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Introduction to Tangent Graph

Special Angles and Their Exact Values

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values derived from special triangles. For 30°, these values come from the 30°-60°-90° triangle, enabling precise calculation without a calculator.
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45-45-90 Triangles

Using the 30°-60°-90° Triangle

The 30°-60°-90° triangle has side ratios of 1:√3:2. Knowing these ratios allows direct computation of trigonometric functions for 30°, such as tan 30° = opposite/adjacent = 1/√3, which can be rationalized to √3/3.
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30-60-90 Triangles
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