Finding all values of $\theta$ which describes a straight line

Question

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed through. The question is below, I shall keep the question updated shortly with my attempts.

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Do note that I haven't attempted Question (b) yet, I would prefer if clear hints are given to work (a) out first so that I can attempt (b) further on.

Attempts so far

Thanks to fellow Maths.SE users.

$$1-2\cos\theta = 0$$ $$\Rightarrow -2cos\theta = -1$$ $$\Rightarrow \cos\theta = \frac{1}{2}$$ $$\Rightarrow \theta = \cos^{-1}(\frac{1}{2}) = 60$$

Answer 1

(a) Fucntion $y$ describes strainght line iff coefficient of $x^2$ is $0$, so $$1-2\cos \theta=0\implies\cos\theta=\frac12$$ From this we get $$\sin \theta =\pm\sqrt{1-\cos^2\theta}=\pm\frac{\sqrt3}2$$ So, equations of these lines are $$y=\pm\frac{\sqrt3}2x+\frac12$$ (b) Let $y=0$. Discriminant of quadratic equation for $x$ is $$D=\sin^2\theta+8\cos^2\theta-4\cos\theta$$ And $$\frac{dD}{d\theta}=4\sin\theta-14\sin\theta\cos\theta$$ Solving $4\sin\theta-14\sin\theta\cos\theta=0$ you will get that minimum value of $D$ is at $\theta=\cos^{-1}\frac27+2k\pi$ for all $k\in\mathbb{Z}$. At these points $D=\frac37$.

Answer 2

Assuming the variable in your equation is $x$ and $\theta$ is a fixed real number (which the statement of your problem fails to specify), then you're looking to get only first or zeroth powers of $x$, as in :

$y = ax + b, (a,b) \in \mathbb{R}^2$

where $ax$ is the first-power term and $b = bx^0$ the zeroth-power term.

This means that terms of higher power ($x^2$, $x^3$, ... or indeed any term of the form $x^n, n \in [[2, +\infty]]$) need to disappear. Which are these terms in your example ? How can you go about making them be equal zero ?