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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

Which of the following expressions is equivalent to $left parenthesis sine 24 degree right parenthesis left parenthesis cosine 66 degree right parenthesis plus left parenthesis cosine 24 degree right parenthesis left parenthesis sine 66 degree right parenthesis$ ?

$2 left parenthesis cosine 66 degree right parenthesis left parenthesis sine 24 degree right parenthesis$

$2 left parenthesis cosine 66 degree right parenthesis plus 2 left parenthesis cosine 24 degree right parenthesis$

$left parenthesis cosine 66 degree right parenthesis squared plus left parenthesis cosine 24 degree right parenthesis squared$

$left parenthesis cosine 66 degree right parenthesis squared plus left parenthesis sine 24 degree right parenthesis squared$

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Explanation

Choice C is correct. The sine of an angle is equal to the cosine of its complementary angle. Since angles with measures $24 degree$ and $66 degree$ are complementary to each other, $sine 24 degree$ is equal to $cosine 66 degree$ and $sine 66 degree$ is equal to $cosine 24 degree$ . Substituting $cosine 66 degree$ for $sine 24 degree$ and $cosine 24 degree$ for $sine 66 degree$ in the given expression yields $left parenthesis cosine 66 degree right parenthesis left parenthesis cosine 66 degree right parenthesis plus left parenthesis cosine 24 degree right parenthesis left parenthesis cosine 24 degree right parenthesis$ , or $left parenthesis cosine 66 degree right parenthesis squared plus left parenthesis cosine 24 degree right parenthesis squared$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.