In Exercises 54–57, solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. A = 22.3°, c = 10

Verified step by step guidance

1. Identify the given elements of the right triangle: angle \(A = 22.3^\circ\) and hypotenuse \(c = 10\).
2. Recall that in a right triangle, the sides are related to the angles by the sine, cosine, and tangent functions. Since \(c\) is the hypotenuse, use the definitions: \(\sin A = \frac{a}{c}\) and \(\cos A = \frac{b}{c}\), where \(a\) and \(b\) are the legs opposite and adjacent to angle \(A\), respectively.
3. Calculate side \(a\) (opposite to angle \(A\)) using the sine function: \(a = c \times \sin A = 10 \times \sin 22.3^\circ\).
4. Calculate side \(b\) (adjacent to angle \(A\)) using the cosine function: \(b = c \times \cos A = 10 \times \cos 22.3^\circ\).
5. Find the remaining angle \(B\) by using the fact that the sum of angles in a triangle is \(180^\circ\), and one angle is \(90^\circ\): \(B = 90^\circ - A = 90^\circ - 22.3^\circ\).

Key Concepts

Right Triangle Properties

A right triangle has one angle of 90°, and the other two angles sum to 90°. Knowing one acute angle and one side allows the use of trigonometric ratios to find the remaining sides and angles.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. For example, sine of an angle equals the opposite side over the hypotenuse, which helps solve for unknown sides or angles.

Rounding and Angle Measurement

Final answers for side lengths should be rounded to two decimal places, and angles should be expressed to the nearest tenth of a degree. This ensures precision and clarity in the solution.