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

Verified step by step guidance

1. Identify the two given sides of the right triangle. Label the sides as opposite (O), adjacent (A), or 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 it is not given. This step is essential to have all three sides before finding the angles.
3. Apply the appropriate trigonometric ratio to find one of the non-right angles. For example, use sine: \(\sin \theta = \frac{O}{H}\), cosine: \(\cos \theta = \frac{A}{H}\), or tangent: \(\tan \theta = \frac{O}{A}\) depending on the sides you know.
4. Calculate the angle \(\theta\) by taking the inverse trigonometric function (arcsin, arccos, or arctan) of the ratio found in the previous step. This will give the angle in degrees.
5. Find the other non-right angle by subtracting the first angle from 90 degrees, since the sum of angles in a right triangle is 90 degrees (excluding the right angle). Convert the decimal degrees to degrees and minutes for the final answer.

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 relate 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 to find angles.

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 engineering.