Question

Find the exact value of arccos (sin (pi divided by six)). For full credit, explain your reasoning.

Answer 1

I need help explaining. I have the answer, but I need to get this question correct!

Answer 2

Lemme explain something really quick in regards to trig functions.

Answer 3

If I were to ask you for \(\Large\rm \cos\theta\),
you would know that it's 3/5.

But more specifically, when taking the trig function of an angle, we're `solving for a numerical value`, the ratio of sides.

If I asked for \(\Large\rm \arccos\dfrac{3}{5}\),
This gives us back our angle \(\Large\rm \theta\).

So when you take the inverse trig function of something, you're `solving for an angle`.

Answer 4

So with that in mind,\[\Large\rm \arccos\left(\sin \frac{\pi}{6}\right)\]What does sin(pi/6) give you?

Answer 5

1/2

Answer 6

\[\Large\rm \arccos\left(\frac{1}{2}\right)\]Mmmm ok good.
Since we're taking the inverse trig function of this value, we're solving for an angle yes?

\[\Large\rm \arccos\left(\frac{1}{2}\right)=\theta\]

Answer 7

Do you remember how to switch between ... like cosine and arccosine?
You just switch the arguments.\[\Large\rm \cos \theta=(stuff) \qquad\to\qquad \arccos(stuff)=\theta\]

Answer 8

yeah

Answer 9

So we can write our inverse function back in terms of cosine.
\[\Large\rm \cos \theta=\frac{1}{2}\]
Do you remember which special angle gives you 1/2?

Answer 10

30

Answer 11

I made a tiny boo boo I guess.
Since we `started with` the inverse cosine function,
we have to remember that our output is restricted to quadrants 1 and 2.

Ummm I think it's 60 isn't it?
It turns out that 300 also gives us 1/2 but that is in quadrant 4 so that one doesn't work for us.

Answer 12

Are you talking about in the Unit Circle? Cause the sin (π/6) gives us 1/2 which is the y value of 30 degrees the Unit Circle

Answer 13

Correct, but now we're dealing with cosine, not sine.

Answer 14

The inner function was sin(30) which gave us 1/2.
Now we're trying to figure out what angle when we take the cosine of it, gives us 1/2.

It's not 30 again. :o

Answer 15

It's 60. :p now that I check my Unit Circle

Answer 16

Ah yes :) good!