Find angle between 0 and pi that's coterminal with 17pi/6?

**zoiberg137** · September 2nd 2012, 05:51 AM

I know the answer is 5pi/6.

I just have no idea how to get that answer!

I understand coterminals when starting with a degree...add or subtract 360 to get between 0 and 360. That makes sense to me. But how do i find it when starting with a radian? Please Explain!!

Thank you!

**Prove It** · September 2nd 2012, 05:58 AM

Originally Posted by zoiberg137

I know the answer is 5pi/6.

I just have no idea how to get that answer!

I understand coterminals when starting with a degree...add or subtract 360 to get between 0 and 360. That makes sense to me. But how do i find it when starting with a radian? Please Explain!!

Thank you!

Well, $\displaystyle \begin{align*} 360^{\circ} = 2\pi^{C} \end{align*}$ , so to find the coterminal angles are found by adding or subtracting multiples of $\displaystyle \begin{align*} 2\pi \end{align*}$ .

**zoiberg137** · September 2nd 2012, 06:08 AM

2pi to the square root of C???? What is C??? havent seen that yet...

Ok so I think I kind of understand how to subtract 2pi and come out with 5pi/6. But then how do I know thats between 0 and 2pi?? How do I know just from looking at it that the original problem, 17pi/6, isn't already between 0 and pi?

I feel like I have totally missed something here...I swear my book does not explain this lol.

**Prove It** · September 2nd 2012, 06:08 AM

Originally Posted by zoiberg137

2pi to the square root of C???? What is C??? havent seen that yet...

Ok so I think I kind of understand how to subtract 2pi and come out with 5pi/6. But then how do I know thats between 0 and 2pi?? How do I know just from looking at it that the original problem, 17pi/6, isn't already between 0 and pi?

I feel like I have totally missed something here...I swear my book does not explain this lol.

C is the symbol for radians (literally means "number of lengths of the radius on the Circumference").

As for knowing if it's between 0 and $\displaystyle \begin{align*} 2\pi \end{align*}$ , multiply $\displaystyle \begin{align*} 2\pi \end{align*}$ by the denominator (6) so that it looks like $\displaystyle \begin{align*} 2\pi = \frac{12\pi}{6} \end{align*}$ . You can clearly see that $\displaystyle \begin{align*} \frac{17\pi}{6} > \frac{12\pi}{6} \end{align*}$ .

**Plato** · September 2nd 2012, 08:36 AM

Originally Posted by zoiberg137

I know the answer is 5pi/6.
I just have no idea how to get that answer!
I understand coterminals when starting with a degree...add or subtract 360 to get between 0 and 360. That makes sense to me. But how do i find it when starting with a radian? Please Explain!

Here is the way I think about.
If $0\le\theta<2\pi$ then adding $2k\pi,~k\in\mathbb{Z},$ gives an equivalent angle became each additional $2\pi$ is a complete rotation.

Thus $\frac{17\pi}{6}=\frac{5\pi}{6}+2\pi.$

**Deveno** · September 2nd 2012, 09:20 AM

how do we know which multiple of 2π to add/subtract?

we want:

0 ≤ 17π/6 + 2πk < 2π

multiply by 6:

0 ≤ 17π + 12πk < 12π

0 ≤ (17 + 12k)π < 12π

multiply by 1/π:

0 ≤ 12k + 17 < 12

subtract 17:

-17 ≤ 12k < -5

multiply by 1/12:

-17/12 ≤ k < -5/12

since k is an integer between -5/12 and -17/12, it must be that k = -1.

therefore the angle we want is:

17π/6 - 2π = 17π/6 - 12π/6 = 5π/6

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