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

Verified step by step guidance

1. Identify the given right triangle with sides 28, 45, and hypotenuse 53.
2. Recognize that the angle \( \beta \) is opposite the side of length 28 and adjacent to the side of length 45.
3. Use the sine function definition: \( \sin \beta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{28}{53} \).
4. Apply the double angle identity for sine: \( \sin 2\beta = 2 \sin \beta \cos \beta \).
5. Find \( \cos \beta \) using \( \cos \beta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{45}{53} \) and substitute into the double angle identity.

Key Concepts

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In a right triangle, the sine, cosine, and tangent functions are defined as the ratios of the lengths of the opposite side to the hypotenuse, adjacent side to the hypotenuse, and opposite side to the adjacent side, respectively. For angle β in the given triangle, the sine can be calculated as sin(β) = opposite/hypotenuse = 28/53.

Double Angle Formulas

Double angle formulas are used to express trigonometric functions of double angles in terms of single angles. For sine, the formula is sin(2θ) = 2sin(θ)cos(θ). This means that to find sin(2β), we first need to determine sin(β) and cos(β) using the triangle's side lengths, and then apply the formula to find the exact value.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem can be used to verify the side lengths of the triangle and to find missing lengths. In this case, it confirms that 28² + 45² = 53², ensuring the triangle is valid for trigonometric calculations.