Question

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!

Answer 1

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


So if `costheta = 1`, then `theta = 0`. Thus `theta != pi/2, (3pi)/2, pi/6, ...`, 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)`

`4sinthetacostheta + ((2sin^2theta)(1 -tan^2theta))/(2tantheta)`

`4sinthetacostheta + (2sin^2theta - 2sin^2thetatan^2theta)/(2tantheta)`

`4sinthetacostheta+ sin thetacostheta- sin^2thetatantheta`

`5sinthetacostheta - sin^3theta/costheta`

A lot of expressions cancel to things like `secx` or `tan(2x)` 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 `sinalpha = 1/sqrt(32)`. Since `cscalpha = 1/sinalpha`, we can see that `cscalpha = sqrt(32)`.

Now from above you can see that `1 + cot^2x = csc^2x`.

`1 +cot^2alpha = 32`

`cot^2alpha = 31`

`cotalpha= +-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 `+-`.

Hopefully this helps, please ask if you have any further questions!