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 = 30.4, c = 50.2

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Identify the sides and angles in the right triangle. Here, side \(a\) corresponds to \(p\), side \(c\) corresponds to \(r\) (the hypotenuse), and side \(b\) corresponds to \(q\). Given: \(a = 30.4\) and \(c = 50.2\).

Use the Pythagorean theorem to find the missing side \(b\) (or \(q\)):

\[b = \sqrt{c^2 - a^2} = \sqrt{50.2^2 - 30.4^2}\]

Calculate angle \(Q\) using the sine function, since \(\sin(Q) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}\):

\[Q = \sin^{-1}\left(\frac{a}{c}\right) = \sin^{-1}\left(\frac{30.4}{50.2}\right)\]

Calculate angle \(P\) by subtracting angle \(Q\) from 90 degrees (since the triangle is right-angled):

\[P = 90^\circ - Q\]

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Right Triangle Properties

A right triangle has one angle of 90 degrees, and the side opposite this angle is the hypotenuse, the longest side. The other two sides are called legs. Understanding these properties helps in applying trigonometric ratios and the Pythagorean theorem to solve for unknown sides or angles.

Pythagorean Theorem

This theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. It is expressed as r² = p² + q². This relationship is essential for finding missing side lengths when two sides are known.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Sine = opposite/hypotenuse, Cosine = adjacent/hypotenuse, and Tangent = opposite/adjacent. These ratios are used to find unknown angles or sides when some measurements are given.