In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, c = 2, B = 90°

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Identify the type of triangle: Since angle B is 90°, this is a right triangle.

Use the Pythagorean theorem to find the missing side b: \( a^2 + b^2 = c^2 \). Substitute the known values: \( 5^2 + b^2 = 2^2 \).

Solve for \( b^2 \): \( b^2 = 2^2 - 5^2 \).

Calculate the angle A using the sine function: \( \sin(A) = \frac{a}{c} \).

Calculate the angle C using the cosine function: \( \cos(C) = \frac{b}{c} \).

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Key Concepts

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

Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant. This law is particularly useful for solving triangles when given two angles and a side or two sides and a non-included angle.

Right Triangle Properties

In a right triangle, one angle measures 90 degrees, which allows the use of specific trigonometric ratios: sine, cosine, and tangent. These ratios relate the angles to the lengths of the sides, making it easier to find unknown side lengths or angles. For example, in a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.

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 is essential for finding the length of a side when two sides are known, and it is a fundamental principle in trigonometry that underpins many calculations involving right triangles.

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