Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. b = 2, c = 7

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Identify the sides and angles in the right triangle. Here, side \(b\) corresponds to \(p = 2\), side \(c\) corresponds to \(r = 7\), and side \(q\) is unknown. Angle \(R\) is the right angle (90°).

Use the Pythagorean theorem to find the length of side \(q\). The formula is \(q = \sqrt{r^2 - p^2}\), where \(r\) is the hypotenuse and \(p\) is one leg of the triangle.

Calculate angle \(Q\) using the sine function: \(\sin(Q) = \frac{p}{r}\). Then find \(Q\) by taking the inverse sine: \(Q = \sin^{-1}\left(\frac{p}{r}\right)\).

Calculate angle \(P\) by subtracting angle \(Q\) from 90°, since the sum of angles in a right triangle is 180° and one angle is 90°. So, \(P = 90^\circ - Q\).

Round the lengths and angles to the required precision: lengths to two decimal places and angles to the nearest tenth of a degree.

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Key Concepts

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

Right Triangle Properties

A right triangle has one angle of 90 degrees. The side opposite this angle is the hypotenuse, the longest side. The other two sides are called legs. Understanding these properties helps identify which sides and angles to use in calculations.

Pythagorean Theorem

This theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²). It is essential for finding the length of an unknown side when the other two sides are known.

Trigonometric Ratios (Sine, Cosine, Tangent)

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

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