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. Rearrange the formula to solve for the unknown side.
3. Choose the appropriate trigonometric ratio to find the unknown angle: 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: \(\theta = \sin^{-1}(\frac{O}{H})\), \(\theta = \cos^{-1}(\frac{A}{H})\), or \(\theta = \tan^{-1}(\frac{O}{A})\).
5. Convert the decimal degree result into 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 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.

Conversion of Decimal Degrees to Degrees and Minutes

Angles are often expressed in degrees and minutes, where one degree equals 60 minutes. After calculating an angle in decimal degrees, converting the fractional part into minutes provides a more precise and conventional angle measurement.