If \( \cos \angle B=\sin \angle A \) and \( m \angle A=32^{\circ} \), what is the measure of \( \angle B \) ? \( 32^{\circ} \) \( 58^{\circ} \) \( 90^{\circ} \) \( 148^{\circ} \)

To solve for \( \angle B \), we start with the equation given: \[ \cos \angle B = \sin \angle A \] We know that \( \angle A = 32^{\circ} \), so we can substitute: \[ \cos \angle B = \sin(32^{\circ}) \] Using the co-function identity, we recognize that: \[ \sin(32^{\circ}) = \cos(90^{\circ} - 32^{\circ}) = \cos(58^{\circ}) \] Thus, we have: \[ \cos \angle B = \cos(58^{\circ}) \] Since cosine is positive in the first and fourth quadrants, the solutions for \( \angle B \) could be: \[ \angle B = 58^{\circ} \quad \text{or} \quad \angle B = 360^{\circ} - 58^{\circ} \] However, the second option results in \( 302^{\circ} \), which is not among the provided options. Therefore, the measure of \( \angle B \) is: \[ \boxed{58^{\circ}} \] Historical Background: The sine and cosine functions have origins dating back thousands of years to ancient civilizations. The Greeks, particularly Hipparchus, were among the first to systematically study these functions for understanding astronomy and navigation, paving the way for modern trigonometry! Real-World Application: Trigonometric functions are essential in various fields, from engineering to music. For instance, in architecture, cosine and sine functions help calculate forces and angles in structures, ensuring they withstand environmental stress, keeping our buildings safe and sturdy!