Question

Prove the following trig identity

Answer 1

\[\frac{ 2 }{ \sqrt{3}cosx+sinx } = \sec(\pi/6 - x)\]

Answer 2

So... where would I start? I know the formula for cos(a - b) but not sec(a - b)... is there a way to derive it?

I have 4 boxes to fill in (four steps).

Answer 3

sure sec(a-b) should equal 1 / cos(a-b)

Answer 4

so that would be

Answer 5

\[\frac{ 1 }{ \cos(\pi/6)*\cos(x)+\sin(\pi/6)*\sin(x) }\]

Answer 6

Oh! Okay :)

Answer 7

Can we start with the left side for this problem though? I'm supposed to start with the side that looks the most complicated.

Answer 8

well we can but the actually deriving will come from the right side.

Answer 9

basically divide the numerator and denominator by 2

Answer 10

and you will have the 1 over equation with the cos and sine of pi/6 in its exact value.

Answer 11

\[\frac{ 1 }{ \sqrt 3 / 2 \cos(x) + 1/2 \sin(x)}\]

Answer 12

I'm sorry... I'm a little lost :P

Answer 13

So I would divide by 2, then do what?

Answer 14

\[\large\rm \frac{1}{\frac{\sqrt3}{2}\cos x + \frac12\sin x}\]Factor a 1/2 out of each term,\[\large\rm \frac{1}{(\sqrt3\cos x + \sin x)\frac12}\]ya? :)
and thennnnn.....?
I think you see it :D

Answer 15

what is the cos of pi/6 and sine of pi/6

Answer 16

in exact form

Answer 17

Ahhh looks like Pebbles ran off to eat some almond butter -_- again

Answer 18

Ahh! Sorry!
Was doing some engineering homework...
hmmm. I don't see it D:

Answer 19

I love that nickname tho :D

Answer 20

You were ok with the cos(pi/6) and sine though?

Answer 21

\[\large\rm \frac{1}{(\sqrt3\cos x + \sin x)\frac12}\quad=\quad \frac{1}{(\sqrt3\cos x + \sin x)}\cdot\frac{1}{\left(\frac12\right)}\]

Answer 22

cos(pi/6) = sqrt3/2
sin(pi/6) = 1/2

The first fraction becomes 1/(sqrt3 + 1)

Answer 23

And then.... hmm...

Answer 24

\[\large\rm \sec\left(\frac{\pi}{6}-x\right)\]secant to cosine,\[\large\rm \frac{1}{\cos\left(\frac{\pi}{6}-x\right)}\]Cosine Angle Difference Formula,\[\large\rm \frac{1}{\cos\frac{\pi}{6}\cos x+\sin\frac{\pi}{6}\sin x}\]Then simplifying our cos(pi/6) and sin(pi/6),\[\large\rm \frac{1}{\frac{\sqrt3}{2}\cos x+\frac12\sin x}\]Do you understand up to that point?

Answer 25

That is perfect! I understand exactly what you did. And from there, just multiply by 2/2 yeah?

Awesome. So starting with the right side is definitely easier... I think I'll just stick with that.

Answer 26

Zep, you are my hero! Ze best in ze world.

Answer 27

Woah woah, leaves the puns to the big boys -_-
Nahhh I guess that was pretty good

Answer 28

Starting with the right side is definitely easier - even though normally you start with the more complex-looking side, that rule fails when one side has an angle sum or difference involved.

I just wanted to be a part of this, since I didn't get a chance to earlier :(

Answer 29

Agent, haha. I have plenty more questions if you're feeling left out :)

Answer 30

Tag me sometimes ;P