1978 AHSME Problems/Problem 15

Problem 15

If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is

$\textbf{(A) }-\frac{4}{3}\qquad \textbf{(B) }-\frac{3}{4}\qquad \textbf{(C) }\frac{3}{4}\qquad \textbf{(D) }\frac{4}{3}\qquad\\ \textbf{(E) }\text{not completely determined by the given information}$

Solution

Squaring the equation, we get $$\sin ^2 x + \cos ^2 x + 2 \sin x \cos x = \frac{1}{25} =\Rrightarrow 2\sin x \cos x = -\frac{24}{25} \Rrightarrow \sin x \cos x = -\frac{12}{25}$$ Recall that $(a-b)^2 = (a+b)^2 - 4ab$, so $$(\sin x - \cos x)^2 = (\sin x + \cos x)^2 - 4 \sin x \cos x$$ $$(\sin x - \cos x)^2 = \frac{1}{25} - 4 (- \frac{12}{25})$$ $$(\sin x - \cos x) = \frac{7}{5}$$ We can now solve for the values of $\sin x$ and $\cos x$ $$\sin x + \cos x = \frac{1}{5}$$ $$\sin x - \cos x = \frac{7}{5}$$ $$2 \sin x = \frac{8}{5} \Rrightarrow \sin x = \frac{4}{5}, \cos x = -\frac{3}{5}$$ Since $\tan x = \frac{\sin x}{\cos x}$, we have $$\tan x = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}$$ $\boxed{A}$

~JustinLee2017