Problem 82 - Algebra and Trigonometry

Question

Describe one similarity and one difference between the definitions of \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is an acute angle of a right triangle.

Step-by-Step Solution

Answer

Both \(\sin \theta\) and \(\cos \theta\) express a ratio of lengths in a right triangle, but \(\sin \theta\) involves the ratio of the opposite side over the hypotenuse, while \(\cos \theta\) involves the ratio of the adjacent side over the hypotenuse.

Step 1: Understanding Sine

The sine of an angle \(\theta\) in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse. Mapped to a unit circle, \(\sin \theta\) gives the y-coordinate of the arc that subtends angle \(\theta\).

Step 2: Understanding Cosine

The cosine of an angle \(\theta\) is the length of the adjacent side divided by the length of the hypotenuse. Mapped to a unit circle, \(\cos \theta\) gives the x-coordinate of the arc that subtends angle \(\theta\).

Step 3: Detailing Similarities and Differences

Similarity: Both \(\sin \theta\) and \(\cos \theta\) express a ratio of lengths in a right triangle and lie in the range -1 to 1 inclusive. They are also related by the identity \(\sin^2 \theta + \cos^2 \theta = 1\).\n\nDifference: \(\sin \theta\) gives the ratio of the opposite side over the hypotenuse while \(\cos \theta\) gives the ratio of the adjacent side over the hypotenuse. On a unit circle, they also map to different coordinates, sine to the y-coordinate and cosine to the x-coordinate.