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 = 16.8°, b = 30.5

Verified step by step guidance

1. Identify the given information: angle B (which corresponds to angle Q) is 16.8° and side b (which corresponds to side q) is 30.5 units.
2. Since triangle QRP is a right triangle with the right angle at R, use the fact that the sum of angles in a triangle is 180°. Calculate angle P as \(P = 90^\circ - B = 90^\circ - 16.8^\circ\).
3. Use the sine function to find side p (opposite to angle B): \(\sin(B) = \frac{p}{r}\), but since we don't know r yet, use the cosine function with side q: \(\cos(B) = \frac{q}{r}\), rearranged to find \(r = \frac{q}{\cos(B)}\).
4. Once you find r, use the sine function to find p: \(p = r \sin(B)\).
5. Finally, verify your results by checking the Pythagorean theorem: \(p^2 + q^2 = r^2\). Round all lengths to two decimal places and angles to the nearest tenth of a degree.

Key Concepts

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 the relationship between angles and sides in a right triangle is essential for solving unknown lengths or angles.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios allow calculation of unknown sides or angles when one angle and one side are known.

Solving Right Triangles

Solving a right triangle involves finding all unknown sides and angles using given information. This typically requires using trigonometric ratios, the Pythagorean theorem, and angle sum properties. Rounding results appropriately ensures practical and accurate answers.