In Exercises 1–6, use the figures to find the exact value of each trigonometric function.

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

In Exercises 1–6, use the figures to find the exact value of each trigonometric function.sin 2θ