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Geometry and Trigonometry Difficulty: Hard

For two acute angles, $∠ Q$ and $∠ R$ , $\cos (Q) = \sin (R)$ . The measures, in degrees, of $∠ Q$ and $∠ R$ are $x plus 61$ and $4 x plus 4$ , respectively. What is the value of $x$ ?

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Explanation

Choice A is correct. It's given that for two acute angles, $∠ Q$ and $∠ R$ , $\cos (Q) = \sin (R)$ . For two acute angles, if the sine of one angle is equal to the cosine of the other angle, the angles are complementary. It follows that $∠ Q$ and $∠ R$ are complementary. That is, the sum of the measures of the angles is $90$ degrees. It's given that the measure of $∠ Q$ is $x plus 61$ degrees and the measure of $∠ R$ is $4 x plus 4$ degrees. It follows that $(x + 61) + (4 x + 4) = 90$ . By combining like terms, this equation can be rewritten as $5 x + 65 = 90$ . Subtracting $65$ from each side of this equation yields $5 x equals 25$ . Dividing each side of this equation by $5$ yields $x equals 5$ .

Choice B is incorrect. This would be the value of $x$ if $\cos (Q) = \cos (R)$ rather than $\cos (Q) = \sin (R)$ .

Choice C is incorrect. This would be the value of $x$ if $\cos (Q) = - \cos (R)$ rather than $\cos (Q) = \sin (R)$ and if $∠ R$ were obtuse rather than acute.

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