(sin10°+sin20°)/(cos10°+cos20°) equals

1984 AMC 12 Problem 23

Attempt Problem 23 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

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23.

\frac{\sin10^\circ+\sin20^\circ}{\cos10^\circ+\cos20^\circ}

equals

\tan10^\circ+\tan20^\circ

\tan30^\circ

\frac12(\tan10^\circ+\tan20^\circ)

\tan15^\circ

\frac14\tan60^\circ

Apply the sum-to-product identities to both numerator and denominator

Both transformed sums contain the same

\cos5^\circ

factor

Let

R

denote the given ratio. By the sum-to-product identities,

\begin{aligned} R&=\frac{\sin10^\circ+\sin20^\circ} {\cos10^\circ+\cos20^\circ},\\ R&=\frac{2\sin15^\circ\cos5^\circ} {2\cos15^\circ\cos5^\circ}\\ &=\tan15^\circ. \end{aligned}

Therefore, the correct answer is D.