Give the exact value using the Half Angle Formula

[Joe_K]

Homework Statement

Find the exact value, without a calculator using half-angle identities, of sin -7pi/8

Homework Equations

Half-angle formula of sin: sin(x/2) = +/- sqrt (1-cosx)/(2)

The Attempt at a Solution

I am confused as to how to use this formula. So far I have:

sin (-7pi/8) = sqrt (1-cos (-7pi/4))/2

 

[Dick]

That seems like a good start. You'll have to figure out the +/- by hand by figuring out what quadrant -7pi/8 is in.

 

[Joe_K]

From what I figured, -7pi/8 would be in the 3rd quadrant, so I will be taking the negative root assuming I am correct. Am I supposed to take the cos value (x-value) of the point of the terminal ray of the angle -7pi/8? I am unsure how I am supposed to find that value to plug into the formula.

 

[Dick]

From what I figured, -7pi/8 would be in the 3rd quadrant, so I will be taking the negative root assuming I am correct. Am I supposed to take the cos value (x-value) of the point of the terminal ray of the angle -7pi/8? I am unsure how I am supposed to find that value to plug into the formula.

Right on the negative. But all you have to do to finish it is find cos(-7pi/4). That's easy, yes? It's a multiple of 45 degrees.

 

[Joe_K]

Ah, so I will be using the value of sqrt(2)/2 as the value of cos -7pi/8? Since -7pi/4 falls on points (sqrt2/2 [x], sqrt2/2, [y]) on the unit circle.

 

[Dick]

BTW you've got the parentheses wrong on sin (-7pi/8) = sqrt (1-cos (-7pi/4))/2, right? It should be sin (-7pi/8) = sqrt ((1-cos (-7pi/4))/2). The /2 is inside the sqrt.

 

[Dick]

Ah, so I will be using the value of sqrt(2)/2 as the value of cos -7pi/8? Since -7pi/4 falls on points (sqrt2/2 [x], sqrt2/2, [y]) on the unit circle.

Yes, sqrt(2)/2. But that's cos(-7pi/4), right? Not cos(-7pi/8). Let's not hash this up completely.

 

[Joe_K]

Sorry, I typed that incorrectly about the sqrt(2)/2 being the value of cos -7pi/8, when it is really the value of cos -7pi/4.

So, after working through the problem, I ended up with:

sin -7pi/8= - sqrt(1-(sqrt(2)/2)/2)

Which I believe simplifies to:

sin -7pi/8= - sqrt(1-sqrt(2)/4) as the final answer?

 

[Dick]

You've got all the right ideas. But I've got -sqrt((1-sqrt(2)/2)/2). That's just a LITTLE bit different. Try this. Get a calculator and punch in sin(-7pi/8) and then put in your answer. Then figure out where you fluffed a parenthesis. I've been known to do that, just to make sure I'm right.

 

[Joe_K]

Ok, I see where my mistake was now. When I used you solution, it matched what my calculator displayed, which is -.38...

Perhaps simplest form would be something like:

-sqrt((1/2)-(sqrt(2)/4)

but I am not too sure on that. By the way, thank you very much for your help, I appreciate you taking the time to help me.

 

[Dick]

Ok, I see where my mistake was now. When I used you solution, it matched what my calculator displayed, which is -.38...

Perhaps simplest form would be something like:

-sqrt((1/2)-(sqrt(2)/4)

but I am not too sure on that. By the way, thank you very much for your help, I appreciate you taking the time to help me.

You're welcome. And I'm really sure that's right. My 'calculator' says so, even though you left out the last parenthesis. Tricky aren't they?