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' and 'b', where 'a' is the length of one leg and 'b' is the length of the other leg.
2. Step 2: Use the Pythagorean theorem to find the hypotenuse 'c'. The formula is: c = \sqrt{a^2 + b^2}.
3. Step 3: Calculate one of the non-right angles using the tangent function. For example, if you want to find angle $\theta$, use $\tan (\theta )=\frac{a}{b}$. Then, find $\theta$ by taking the arctangent: $\theta ={\tan }^{-1}(\frac{a}{b})$.
4. Step 4: Convert the angle $\theta$ from decimal degrees to degrees and minutes. To do this, take the decimal part of the angle and multiply it by 60 to get the minutes.
5. Step 5: Find the other non-right angle by subtracting $\theta$ from 90 degrees, since the sum of angles in a triangle is 180 degrees and one angle is 90 degrees.

Key Concepts

Right Triangle Properties

A right triangle is defined by one angle measuring 90 degrees. The relationships between the sides and angles are governed by the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Understanding these properties is essential for solving right triangles, as they provide the foundational relationships needed to find unknown angles and lengths.

Trigonometric Ratios

Trigonometric ratios, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. For a right triangle, sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. These ratios are crucial for calculating unknown angles when two sides are known.

Angle Measurement in Degrees and Minutes

Angles can be measured in degrees, where one full rotation is 360 degrees, and in minutes, where one degree is divided into 60 minutes. This method of measurement is particularly useful in trigonometry for providing precise angle values. When solving triangles, it is important to convert between these units as needed, especially when presenting answers in degrees and minutes.