How does the ratio $r=\frac{\sin\alpha}{\cos\theta}$ change as we increase $\theta$ with the constraint $\theta+\alpha=\text{constant}$?

Question

I want to check how does the ratio $r=\dfrac{\sin\alpha}{\cos\theta}$ change as we increase $\theta$ with the constraint $\theta+\alpha=\text{constant}$. What is the best way to check that the ratio decreases as we increase $\theta$ keeping $\theta+\alpha=\text{constant}$.

What is the best way to check whether the ratio increases or decreases without putting values for $\theta,\alpha$? I couldn't make any conclusion by calculating $\dfrac{dr}{d\theta}$.

Answer 1

Let $\theta+\alpha=k$, where $k$ is a constant. Then $$r=\frac{\sin(k-\theta)}{\cos\theta}\,.$$ So $$\frac{dr}{d\theta}=\frac{-\cos(k-\theta)\cos\theta+\sin\theta\sin(k-\theta)}{\cos^2\theta}=\frac{-\cos(k-\theta+\theta)}{\cos^2\theta}=-\frac{\cos k}{\cos^2\theta}$$ The denominator is always positive. Whether $r$ increases or decreases as $\theta$ varies depends on the sign of $\cos k$.

Answer 2

Let $c=\theta+\alpha$, $r=\frac{\sin\alpha}{\cos\theta}=\frac{\sin(c-\theta)}{\cos\theta}=\frac{\sin c\cos \theta-\cos c\sin \theta}{\cos\theta}=\sin c-\cos c\tan\theta$.

Thus $r$ is a function of $\theta$, you are invited to plot a graph of the function. Notice the graph depends on the sign of $\cos c$.

However, the tangent function is not continuous everywhere, so you can't say that the function is decreasing or increasing on $R$. You can only say that the function is decreasing/increasing in an interval.