# What should I study to know how to solve?

## I'm studying for a test, and they have several questions I've never dealt with, and I don't even know what the topics I should study are. Here are some of them: cos δ = 1, which of the following could NOT be a value of δ? Which of the following is equivalent to 2sin2θ+(1–cos2θ)/(tan2θ)? If sin∂=sqrt32, what is the value of cot∂? All that I want to know is WHAT I SHOULD STUDY to know how to solve these type of questions, since I have no idea how to even tackle them. I don't actually need the answers, just the names of the methods. Thanks!

Apr 9, 2018

For the first question, know your unit circle and special angles. Here is an image:

So if $\cos \theta = 1$, then $\theta = 0$. Thus $\theta \ne \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{\pi}{6} , \ldots$, many answers possible.

For the second, you need to know your trig identities. Here is a picture of those that I think are most necessary to learn.

We can simplify as

2(2sinthetacostheta) + (1 -(1 - 2sin^2theta))/((tan theta + tan theta)/(1 - tanthetatantheta)

$4 \sin \theta \cos \theta + \frac{\left(2 {\sin}^{2} \theta\right) \left(1 - {\tan}^{2} \theta\right)}{2 \tan \theta}$

$4 \sin \theta \cos \theta + \frac{2 {\sin}^{2} \theta - 2 {\sin}^{2} \theta {\tan}^{2} \theta}{2 \tan \theta}$

$4 \sin \theta \cos \theta + \sin \theta \cos \theta - {\sin}^{2} \theta \tan \theta$

$5 \sin \theta \cos \theta - {\sin}^{3} \frac{\theta}{\cos} \theta$

A lot of expressions cancel to things like $\sec x$ or $\tan \left(2 x\right)$ which is always really nice.

As for the last problem, this example is implausible as -1 ≤ sin alpha ≤ 1 and $\sqrt{32} > 1$. So I will use $\sin \alpha = \frac{1}{\sqrt{32}}$. Since $\csc \alpha = \frac{1}{\sin} \alpha$, we can see that $\csc \alpha = \sqrt{32}$.

Now from above you can see that $1 + {\cot}^{2} x = {\csc}^{2} x$.

$1 + {\cot}^{2} \alpha = 32$

${\cot}^{2} \alpha = 31$

$\cot \alpha = \pm \sqrt{31}$

If they clarify that $\alpha$ is in quadrant $1$ we can guarantee that it will be positive. Similarly if $\alpha$ is in quadrant $2$ then it will be negative. But when unspecified, keep the $\pm$.