evaluate cos^-1(sqrt(2)sinπ/4)

Question

zepdrix · Answer

$$\large \arccos\left[\sqrt2 \color{royalblue}{\sin\left(\frac{\pi}{4}\right)}\right]$$
So let's work from the inside outward.
Start with the blue part. Do you remember what this special angle produces?

anonymous · Answer

No clue :/

zepdrix · Answer

Booooooo!

zepdrix · Answer

If you're working on problems like this, you're suppose to know that by now :c
Try to take some time and commit the special angles and measurements on the unit circle to memory.

anonymous · Answer

So would the answer for the blue part be a angle in degrees?

anonymous · Answer

Oh never mind! is it square root(2)/2?

zepdrix · Answer

No, we're taking the sine OF an angle. It will produce a length.

Yes, good. That's the one :)

anonymous · Answer

ok so the blue part is square root (2)/2 whats next?

zepdrix · Answer

$$\large \arccos\left[\sqrt2 \color{royalblue}{\frac{\sqrt2}{2}}\right]$$

zepdrix · Answer

Hmm it looks like that inside part will simplify a bit, doesn't it? :o

anonymous · Answer

2/2 which equals 1?

zepdrix · Answer

$$\large \arccos\left[1\right]$$

zepdrix · Answer

Cool looks like we're almost done! :)

anonymous · Answer

Ok so would you just type that into the calculator to get the answer?

zepdrix · Answer

For this one, it turns out yes you can. It will give you a nice easy answer.
But normally that might not work.

Here is how you should think of it.$$\large \arccos\left[1\right]=\theta$$We can rewrite this as the cosine, like so,$$\large \cos \theta=1$$

What special angle produces 1 in the cosine? :O

anonymous · Answer

360 degrees or 2π?

zepdrix · Answer

Yes, but there is a better value we can use :)
What is 2pi equivalent to?
Spin backward one full spin around the unit circle, where do you land? :O

anonymous · Answer

just π?

zepdrix · Answer

No silly! You only spun around half way! :3

anonymous · Answer

Sorry! 0?

zepdrix · Answer

Yah sounds good c:

$\large \cos \theta=1 \qquad \qquad \rightarrow \qquad \qquad \theta=0$

This is actually a bit important. Because the INVERSE cosine, can't produce an angle of 2pi. It has to produce an angle between 0 and pi.
So it's important that you chose the 0, not the 2pi angle.

anonymous · Answer

zepdrix i love your photo

zepdrix · Answer

haha thanks XD Yours is a little scary <:O hehe

anonymous · Answer

Ok thank you so much again! i have a feeling im gonna be thanking you a lot today haha

zepdrix · Answer

:3