# Thread: trig functions and exact values

1. ## trig functions and exact values

OK so the other day Reckoner gave me some great help with these, but now i can't work out this one. What's really confused me is the $\displaystyle x-pi/3$ in the middle. I know the answer should be over 6 (as in like $\displaystyle pi/6$) but I can't work out how the bit in the middle comes into play. Would you do this...

$\displaystyle sinx-sin*pi/3=1/2 sinx=1/2 + sin*3$
etc.....????????????????????

2. Originally Posted by Stevo_Evo_22
OK so the other day Reckoner gave me some great help with these, but now i can't work out this one. What's really confused me is the $\displaystyle x-pi/3$ in the middle. I know the answer should be over 6 (as in like $\displaystyle pi/6$) but I can't work out how the bit in the middle comes into play. Would you do this...

$\displaystyle sinx-sin*pi/3=1/2 sinx=1/2 + sin*3$
etc.....????????????????????
I have, not even in the slightest, any idea what your question is.

3. lol sorry ok....

If 0 is the same as or less than x, which is the same as or less than 2pi, find:

{x:sin(x-pi/3)=1/2}

4. Let $\displaystyle \theta = x - \frac{\pi}{3}$ for visualization purposes. Then:
$\displaystyle \sin \theta = \frac{1}{2} \: \: \Rightarrow \: \: \theta =\frac{\pi}{6}, \frac{5\pi}{6}$ (you should be familiar with these values)

Now, since $\displaystyle \theta = x - \frac{\pi}{3}$, simply solve for x: $\displaystyle x - \frac{\pi}{3} = \frac{\pi}{6} \qquad x - \frac{\pi}{3} = \frac{5\pi}{6}$

5. Originally Posted by Stevo_Evo_22
lol sorry ok....

If 0 is the same as or less than x, which is the same as or less than 2pi, find:

{x:sin(x-pi/3)=1/2}

$\displaystyle \sin\left(x-\frac{\pi}{3}\right)=\frac{1}{2}$

With the given condition that $\displaystyle x\leq{0}$?

If so,

$\displaystyle x-\frac{\pi}{3}=\frac{\pi}{6}+2\pi{n}\quad\text{wher e }n\in\mathbb{Z}$

So then we have that

$\displaystyle x=\frac{\pi}{2}+2\pi{n}\quad\text{where }n\in\mathbb{Z}$

But $\displaystyle x<0\Rightarrow\frac{\pi}{2}+2\pi{n}$

So this implies that $\displaystyle x=\frac{\pi}{2}+2\pi{n}\quad\text{where }n\in\mathbb{Z^-}$

6. Perfect, thanks!

I was getting a similar answer, but didn't know what to do after i got 5pi/6.

Thanks a million and sorry about all these trig questions but we're on holidays so we can't get help from a teacher (and everyone here is always so helpful)

$\displaystyle \sin\left(x-\frac{\pi}{3}\right)=\frac{1}{2}$

With the given condition that $\displaystyle x\leq{0}$?[quote]

No, where x is between or the same as 0 and 2pi (in a unit circle).

$\displaystyle \sin\left(x-\frac{\pi}{3}\right)=\frac{1}{2}$

With the given condition that $\displaystyle x\leq{0}$?

No, where x is between or the same as 0 and 2pi (in a unit circle).
oh ...sorry...a little communication problem

9. Thats ok, dw bout it. Here's another one that doesn't really make sense:

x is between 0 and 2pi again but find: {x: cos(x-pi/6) = sqrt3/2}

Should the answer be 11pi/6 ?

10. Originally Posted by Stevo_Evo_22
Thats ok, dw bout it. Here's another one that doesn't really make sense:

x is between 0 and 2pi again but find: {x: cos(x-pi/6) = sqrt3/2}

Should the answer be 11pi/6 ?
$\displaystyle \cos \left(x - \frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \ \ \ \ 0 \le x \le 2\pi$

$\displaystyle \cos ^{-1} \left( \frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}, \frac{11\pi}{6}$

But these value satify the equation when $\displaystyle x-6 = \cos \left(x - \frac{\pi}{6}\right)$ and we want the value for $\displaystyle x$ hence add $\displaystyle 6$ to both sides.

$\displaystyle \therefore x= \frac{\pi}{3}, 2\pi$

11. That's the answer i got, but the book's answer said it was wrong! I'm really getting sick of the books wrong answers Thanks heaps!

12. Originally Posted by Stevo_Evo_22
That's the answer i got, but the book's answer said it was wrong! I'm really getting sick of the books wrong answers Thanks heaps!
If you are unsure then just insert your $\displaystyle x$ values back into the equation and see if you get the answer you require.

$\displaystyle x= \frac{\pi}{3} \implies \cos \left(\frac{\pi}{3} - \frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ Therefore this value is correct.

$\displaystyle x= \frac{\pi}{3} \implies \cos \left({2\pi} - \frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ Therefore this value is correct too.