Trigonometric Ratio Sum
Let $ \alpha $ and $ \beta $ be angles such that
\[\frac{\cos \alpha}{\cos \beta} + \frac{\sin \alpha}{\sin \beta} = -1.\]Find all possible values of
\[\frac{\cos^3 \beta}{\cos \alpha} + \frac{\sin^3 \beta}{\sin \alpha}.\]Enter all possible values, separated by commas.
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
Let $ k = \frac{\cos \alpha}{\cos \beta} $. Then $ \frac{\sin \alpha}{\sin \beta} = -k - 1, $ so $ \cos \alpha = k \cos \beta $ and $ \sin \alpha = -(k + 1) \sin \beta $. Substituting into $ \cos^2 \alpha + \sin^2 \alpha = 1, $ we get
\[k^2 \cos^2 \beta + (k + 1)^2 \sin^2 \beta = 1.\]Then $ k^2 \cos^2 \beta + (k + 1)^2 (1 - \cos^2 \beta) = 1, $ which leads to
\[\cos^2 \beta = \frac{k^2 + 2k}{2k + 1}.\]Therefore,
\[\sin^2 \beta = 1 - \cos^2 \beta = \frac{1 - k^2}{2k + 1}.\]Hence,
\begin{align*}
\frac{\cos^3 \beta}{\cos \alpha} + \frac{\sin^3 \beta}{\sin \alpha} &= \cos^2 \beta \cdot \frac{\cos \beta}{\cos \alpha} + \sin^2 \beta \cdot \frac{\sin \beta}{\sin \alpha} \\
&= \frac{k^2 + 2k}{2k + 1} \cdot \frac{1}{k} + \frac{1 - k^2}{2k + 1} \cdot \frac{1}{-k - 1} \\
&= \frac{k + 2}{2k + 1} + \frac{k - 1}{2k + 1} \\
&= \frac{2k + 1}{2k + 1} = \boxed{1}.\end{align*}
Freeflows