Question

PLEASE HELP!!!
Rewrite each trig. expression in simpler form using trigonometric identities.
a) 2sin5pcos5p
b) sin pi/6 cos pi/4 - cos pi/6 sin pi/4

Answer 1

Recall your Sine Double Angle Identity:
\(\large\rm 2\sin(\color{orangered}{\theta})\cos(\color{orangered}{\theta})=\sin(2\color{orangered}{\theta})\)

So if you have,
\(\large\rm 2\sin(\color{orangered}{5p})\cos(\color{orangered}{5p})\)
what will you end up with?

Apply the identity.

Answer 2

Thanks for helping by the way! So, I got sin (2 * 5p) = 2sin5pcos5p

Answer 3

Good good good.
simplifying to sin(10p), ya? :)

Answer 4

Mhm!
And then I divided sin5p from both sides.

Answer 5

So I got sin2p = 2cos5p

Answer 6

What :U
no no, sin(10p) is your solution.
Stop there.

Answer 7

no no no no don't ever do that again XD

Answer 8

ever :P

Answer 9

Oh! That makes a lot of sense. Thank you so much!

Answer 10

sin(10p)/sin(5p) does not equal sin(2p).
The numbers are locked inside the sine operation.
You can't divide like that :)

Answer 11

So, I'd just keep whatever is on the left side?

Answer 12

If you wrote the sin(10p) on the left side, then yes :)
we usually draw our equals signs on the right side, but thats ok hehe

Answer 13

For the other problem, apply your Angle Difference Formula for Sine,
\(\large\rm \sin A\cos B-\sin B\cos A=\sin(A-B)\)

Answer 14

So, then A would equal pi/6 and B would equal pi/4

Answer 15

Would that just simplify to sin (pi/6-pi/4) ?

Answer 16

Yes, but we would like to simplify the angle if we can :)
We'll need a common denominator.

Answer 17

Unless your teacher doesn't care, I dunno :D

Answer 18

The common denominator would be 12.
Then, it's 2pi/12 - 3pi/12, and you'd get -1pi/12?

Answer 19

Mmm k great!

=sin(-pi/12)
Since sine is an `odd function`, the negative can pass outward like this,
=-sin(pi/12)

But that last step isn't too important :) don't worry about it.

Answer 20

Alright! Thanks! Also, if you have the time, would you mind helping me to understand what an odd function is?

Answer 21

It might be easier to first understand what an `even function` is.
You're familiar with a parabola?

Answer 22

Yes!

Answer 23

Even functions are symmetric about the y-axis.
Parabola is a good example, is has the same shape on the left and right side of the y-axis.

But what's more important is that the negative always gets "squared" away in an even function.

So even functions follow this property:
\(\large\rm f(-x)=f(x)\)
Example: If you plug x=-3 into the function,
it should give you the same result as plugging x=3 into the function.

\(\large\rm g(x)=(x)^2\)
At x=2 and x-2,
\(\large\rm g(-2)=(-2)^2=4\)
\(\large\rm g(~~~2)=(~~~2)^2=4\)

Answer 24

Oh! I see. So, an odd function would be like.. a cubic function?

Answer 25

Yes, good good.
\(\large\rm f(x)=(x)^3\)

And you recall what happens when you multiply three negatives together?

Answer 26

Yeah! It turns into a negative.

Answer 27

\(\large\rm f(-2)=(-2)^3=-(2)^3=-f(2)\)
Good good good.
So we can take the negative out front like that.

Answer 28

So odd function follows this property:
\(\large\rm f(-x)=-f(x)\)
the negative can come out in front of the function like that.

Answer 29

And for the even functions, it wouldn't matter if the x value was positive or negative, it would still turn out positive?

Answer 30

Yes, good :)
Positive.

And it turns out that Cosine is an example of an `even function`.
So we could say that \(\large\rm \cos(-x)=\cos(x)\)
The negative can go away.
\(\large\rm \cos(-\pi/3)=\cos(\pi/3)\)

Answer 31

While Tangent and Sine are `odd functions`.

Answer 32

Would that be because cosine represents the x value?

Answer 33

(regarding unit circle coordinates)

Answer 34

Yes yes.
I'm not sure if I can draw it for you since I'm on my laptop right now. Hmm

Answer 35

That's alright, I understand

Answer 36

So ya, they both have the same x-coordinate :)