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Chapter 5: Problem 82

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.

Short Answer

Expert verified
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 by step solution

01

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 \(\theta\).
02

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 \(\theta\).
03

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.

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