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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 1
Capitolo 3, Problema 1

In Exercises 1–6, use the figures to find the exact value of each trigonometric function.
Right triangle with sides labeled 28, 45, and hypotenuse 53, angle beta marked.
sin 2θ

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Identify the sides of the right triangle relative to angle \( \beta \): the opposite side is 28, the adjacent side is 45, and the hypotenuse is 53.
Recall the double-angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Here, \( \theta = \beta \).
Calculate \( \sin \beta \) using the definition of sine: \( \sin \beta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{28}{53} \).
Calculate \( \cos \beta \) using the definition of cosine: \( \cos \beta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{45}{53} \).
Substitute \( \sin \beta \) and \( \cos \beta \) into the double-angle formula: \( \sin 2\beta = 2 \times \frac{28}{53} \times \frac{45}{53} \).

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Right Triangle Trigonometric Ratios

In a right triangle, the primary trigonometric functions—sine, cosine, and tangent—are defined as ratios of the sides relative to an angle. For angle β, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios help find exact values using side lengths.
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Introduction to Trigonometric Functions

Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle, which can be derived from the triangle's side lengths.
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Double Angle Identities

Using Side Lengths to Find Trigonometric Values

Given the side lengths of a right triangle, you can calculate the sine and cosine of an angle by dividing the appropriate sides. For angle β, sin(β) = opposite/hypotenuse = 28/53 and cos(β) = adjacent/hypotenuse = 45/53, which are essential for applying the double-angle formula.
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Finding Missing Side Lengths
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