Refer to the right triangle diagram and the given...

Refer to the right triangle diagram and the given information to solve the right triangle. Write your answers for angle measures in decimal degrees.
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$$\beta=47.2^{\circ} \text { and } a=9.75 \mathrm{mi}$$

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Key Concepts

Right Triangle Properties

A right triangle is defined by having one 90-degree angle, which ensures that the other two angles are complementary, meaning they add up to 90 degrees. This property is fundamental when solving the triangle as it allows the determination of one unknown angle if the other is known, and provides the foundation for applying trigonometric relationships.

Trigonometric Ratios

Trigonometric ratios such as sine, cosine, and tangent relate the angles of a right triangle to its side lengths. These ratios are crucial for solving for unknown sides or angles when one side length and one angle (other than the right angle) are known.

Inverse Trigonometric Functions

When one of the trigonometric ratios is known, inverse trigonometric functions (like arcsine, arccosine, and arctangent) are used to determine the corresponding angle. This is particularly useful in right triangle problems where an angle needs to be calculated from given side lengths.

Pythagorean Theorem

The Pythagorean Theorem establishes a relationship between the lengths of the sides of a right triangle, stating that the sum of the squares of the two legs is equal to the square of the hypotenuse. This theorem is essential for finding an unknown side when the other two sides are known.

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