Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

Verified step by step guidance

1. Step 1: Identify the given sides of the right triangle. Let's denote them as a (adjacent), b (opposite), and c (hypotenuse).
2. Step 2: Use the Pythagorean theorem, $a^2+b^2=c^2$, to verify the relationship between the sides or to find the missing side if only two sides are given.
3. Step 3: Calculate one of the non-right angles using trigonometric ratios. For example, use $\tan (\theta )=\frac{b}{a}$ to find angle $\theta$ or $\sin (\theta )=\frac{b}{c}$ or $\cos (\theta )=\frac{a}{c}$.
4. Step 4: Convert the angle from decimal degrees to degrees and minutes if necessary. This involves taking the decimal part of the degree and multiplying by 60 to get minutes.
5. Step 5: Use the fact that the sum of angles in a triangle is 180 degrees to find the other non-right angle, knowing that one angle is 90 degrees.

Key Concepts

Right Triangle Properties

A right triangle is defined by one angle measuring 90 degrees. The other two angles are complementary, meaning they add up to 90 degrees. The sides of a right triangle are categorized as the opposite side, adjacent side, and hypotenuse, which is the longest side opposite the right angle. Understanding these properties is essential for applying trigonometric functions to solve for unknown angles and sides.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. The primary ratios are sine (sin), cosine (cos), and tangent (tan), defined as follows: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios are fundamental for calculating unknown angles and sides in right triangles when two sides are known.

Angle Measurement in Degrees and Minutes

Angles can be measured in degrees, where one full rotation is 360 degrees. For more precision, angles can also be expressed in degrees and minutes, where one degree is divided into 60 minutes. This format is particularly useful in trigonometry for providing exact angle measures when solving triangles. Understanding how to convert between these measurements is crucial for accurate calculations.