What Is the Equivalent Expression for a Trigonometric Cosine Identity?

[aisha]

I have a question that says an equivalent expression for $$\cos\frac {3\theta} {2} \cos \frac {\theta} {2} + \sin \frac {3\theta} {2} \sin\frac {\theta} {2}$$ is??

How do u get the answer to be $$\cos \theta$$?

Can someone please help me out?

 

[vincentchan]

use addition formulas
$$\cos(x-y)= \cos x \cos y + \sin x \sin y$$

 

[MathStudent]

use the identity
cos(a - b) = cos(a)cos(b) +sin(a)sin(b)

oops: vincentchan beat me to it

 

[aisha]

Ok I know how to write that out but how will it simplify to cos theta, I don't know how to do it?

 

[maverick280857]

Ok I know how to write that out but how will it simplify to cos theta, I don't know how to do it?

Well now that you know the formula for difference, there is no problem at all. Try again. Hint: In $\cos(x-y)= \cos x \cos y + \sin x \sin y$ try and relate x and y to the angles you have been given.

Cheers
Vivek

 

[cepheid]

Ok I know how to write that out but how will it simplify to cos theta, I don't know how to do it?

What you have is what is written on the right hand side in this case, with a = 3theta/2, and b = theta/2. Now, it's an identity...the left hand side and right hand side are "identical", meaning that you can always replace one with the other. So, what happens when you convert the expression you have to the left hand side form?

 

[aisha]

can I convert into degrees and then use special triangles to solve this one?

 

[cepheid]

You don't need to do anything of the sort. This question is really simple! Apply the identity ;)

I'll put the general formula side by side with the specific case. That should make it obvious:

In general, for any x and y:

$$\cos x \cos y + \sin x \sin y = \cos(x-y)$$

Now, in your particular case, you have been given specific values for x and y, but the left side is in *exactly* the same form:

$$\cos\frac {3\theta} {2} \cos \frac {\theta} {2} + \sin \frac {3\theta} {2} \sin\frac {\theta} {2} = ...$$

What can you conclude? Just use the identity,

 

[aisha]

ok i think i get it since cos(x-y) that means that there is nothing left because the theta and beta numbers were the same so all we are left with is theta, I am not really sure how to explain it but i think i get it, I am just started to used identities and its really confusing lol thanks for ur help everyone!

 

[cepheid]

Aisha,

Don't worry...you'll get the hang of identities in no time. Now that you have some inkling of what we were trying to explain, I thought I'd show it explicitly, since you still weren't 100% sure about it.

Here is the trig identity:

$$\cos x \cos y + \sin x \sin y = \cos(x-y)$$

Now, in your particular case, you have been given specific values for x and y:

$$x = \frac {3\theta} {2}$$

$$y = \frac {\theta} {2}$$

So just substitute these values for x and y into the expression above! That's all this question involved:

$$\cos\frac {3\theta} {2} \cos \frac {\theta} {2} + \sin \frac {3\theta} {2} \sin\frac {\theta} {2} = \cos(\frac {3\theta} {2}-\frac {\theta} {2}) = \cos\theta$$

^There's our result.

 

[aisha]

hmmm interesting looks easy but I don't know why its important to know.

 

[dextercioby]

Important to know what?The cosine identity...??If you WANT to know how to solve certain problems,then yes,u can think of it as being important...

Daniel.