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 = 10.8, b = 24.7

검증된 단계별 안내

Identify the sides and angles in the right triangle. Here, side \(p\) is opposite angle \(P\), side \(q\) is adjacent to angle \(P\), and side \(r\) is the hypotenuse opposite the right angle at \(R\).

Use the Pythagorean theorem to find the length of the hypotenuse \(r\). The formula is \(r = \sqrt{p^2 + q^2}\), where \(p = 10.8\) and \(q = 24.7\).

Calculate angle \(P\) using the tangent function, since you know the opposite side \(p\) and adjacent side \(q\). Use \(\tan(P) = \frac{p}{q}\), then find \(P = \arctan\left(\frac{p}{q}\right)\).

Calculate angle \(Q\) by subtracting angle \(P\) from 90 degrees, because the sum of angles in a right triangle is 90 degrees for the two non-right angles: \(Q = 90^\circ - P\).

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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주요 개념

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Pythagorean Theorem

The Pythagorean theorem relates the lengths of the sides in a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This is essential for finding the missing side length when two sides are known.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios define relationships between the angles and sides of a right triangle. Sine, cosine, and tangent functions relate an angle to the ratios of specific sides, enabling calculation of unknown angles or sides when some measurements are given.

Angle Sum Property of Triangles

The sum of the interior angles in any triangle is always 180 degrees. In a right triangle, one angle is 90 degrees, so the other two angles must add up to 90 degrees. This property helps find unknown angles once one non-right angle is known.