Problem understanding value of angles

[babadany2999]

Hello so I am self teaching calculus atm(almost end of calc1) and i've encountered a bunch of problems where i need to know the value of angle x given a trig function like cosx = a , a some number. I've looked for an answer the first time i encountered this but i couldn't find it. So now again i encountered one more problem like this and i need to know all of the values of x where cos x = -1/2, 0 <= x <= 2pi (it's actually an example but again my thinking is not the same as what actually is in the book).
My thinking is that cosine is negative in quadrant 2,3.Since we have cos x = 1/2 at x = pi/3 or at 60* then we should have the negative value(-1/2) at pi/3 + pi/2 (60*+90*) or 5pi/6 . And at pi + pi/3 = 4pi/3 . But in the book it says 2pi/3 and 4pi/3 . I do not understand why this is it...so please help if you can(and sorry for the noob question)

 

[Deleted member 4993]

Hello so I am self teaching calculus atm(almost end of calc1) and i've encountered a bunch of problems where i need to know the value of angle x given a trig function like cosx = a , a some number. I've looked for an answer the first time i encountered this but i couldn't find it. So now again i encountered one more problem like this and i need to know all of the values of x where cos x = -1/2, 0 <= x <= 2pi (it's actually an example but again my thinking is not the same as what actually is in the book).
My thinking is that cosine is negative in quadrant 2,3.Since we have cos x = 1/2 at x = pi/3 or at 60* then we should have the negative value(-1/2) at pi/3 + pi/2 (60*+90*) or 5pi/6 . And at pi + pi/3 = 4pi/3 . But in the book it says 2pi/3 and 4pi/3 . I do not understand why this is it...so please help if you can(and sorry for the noob question)

The formula to use here is:

cos(Θ) = - cos(π ± Θ)​

thus your "answer" is:

Θ = π ± π/3 = 4π/3 or 2π/3

 

[babadany2999]

The formula to use here is:

cos(Θ) = - cos(π ± Θ)​

thus your "answer" is:

Θ = π ± π/3 = 4π/3 or 2π/3

ok but i have a few questions.... first where did this formula come from?i want to obviously understand what i'm doing.secondly what is with the +/- ? and thirdly is it the same for sin as well? meaning sin(x)=-sin(pi+/x) ?
Also thank you very much for the help!

 

[Deleted member 4993]

ok but i have a few questions.... first where did this formula come from?i want to obviously understand what i'm doing.secondly what is with the +/- ? and thirdly is it the same for sin as well? meaning sin(x)=-sin(pi+/x) ?
Also thank you very much for the help!

first where did this formula come from?

These were "exposed" to me, while I was taking a course in Trigonometry. Then those were reinforced (re-introduced) during pre-calculus course. Since you are self-teaching, you may have skipped few of these steps. You can "prove" these formulae through consideration of unit circles or "addition of angles" formulae.

secondly what is with the +/- ?

I do not understand that question. Similar sign is used for solution of quadratic equation. You have certainly seen those. In this context, it is meant to indicate that we have two answers (as shown).

thirdly is it the same for sin as well?

No and Yes. Sine function has similar (not same) relationships:

sin(Θ) = -sin(π + Θ) and -sin(2π - Θ)

There are similar equations for tan functions also. I'll leave those for you to discover (and prove).

 

[babadany2999]

first where did this formula come from?

These were "exposed" to me, while I was taking a course in Trigonometry. Then those were reinforced (re-introduced) during pre-calculus course. Since you are self-teaching, you may have skipped few of these steps. You can "prove" these formulae through consideration of unit circles or "addition of angles" formulae.

secondly what is with the +/- ?

I do not understand that question. Similar sign is used for solution of quadratic equation. You have certainly seen those. In this context, it is meant to indicate that we have two answers (as shown).

thirdly is it the same for sin as well?

No and Yes. Sine function has similar (not same) relationships:

sin(Θ) = -sin(π + Θ) and -sin(2π - Θ)

There are similar equations for tan functions also. I'll leave those for you to discover (and prove).

All righty . Thank you very much brother!