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. Identify the given sides of the right triangle. Label the sides as opposite (O), adjacent (A), and hypotenuse (H) relative to the angle you want to find.
2. Use the Pythagorean theorem \(H^2 = O^2 + A^2\) to find the missing side if only two sides are given and the hypotenuse is not one of them.
3. Apply the appropriate trigonometric ratio to find the unknown angles. For example, use \(\sin \theta = \frac{O}{H}\), \(\cos \theta = \frac{A}{H}\), or \(\tan \theta = \frac{O}{A}\) depending on the sides you have.
4. Calculate the angle in degrees using the inverse trigonometric function, such as \(\theta = \sin^{-1}\left(\frac{O}{H}\right)\), \(\theta = \cos^{-1}\left(\frac{A}{H}\right)\), or \(\theta = \tan^{-1}\left(\frac{O}{A}\right)\).
5. Convert the decimal degrees to degrees and minutes by separating the integer part as degrees and multiplying the decimal part by 60 to get minutes.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem relates the lengths of the sides in a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. It is essential for finding the missing side when two sides are known.

Trigonometric Ratios (Sine, Cosine, Tangent)

Sine, cosine, and tangent ratios connect the angles of a right triangle to the ratios of its sides. These ratios allow calculation of unknown angles or sides when two sides are given, using inverse trigonometric functions for angle determination.

Angle Measurement in Degrees and Minutes

Angles can be expressed in degrees and minutes, where one degree equals 60 minutes. Converting decimal degrees to degrees and minutes is important for precise angle representation, especially in practical applications like navigation or surveying.