Question

Suppose that sin theta = 3/5 and 0 12 years ago

Answer 1

Do you know the formulas for the double angles:

\(\sin 2\theta=2\sin\theta\cos\theta\) and
\(\cos2\theta=2\cos^2\theta-1\)

And also \(\sin^2\theta+\cos^2\theta=1\)

?

Answer 2

I'm sorry I don't know.

Answer 3

But you need these kind of formulas to be able to do anything with this question.
I'm sure you know the last one!

Answer 4

I do know the last one, however I do not know how to use them. I am barely getting to know the material.

Answer 5

After all, we can't use magic, so we have to learn formulas...

Answer 6

OK, starting with the last one: \(\sin^2 \theta=\frac{9}{25}\).
So what is \(\cos^2\theta\) then?

Answer 7

How would I know what to input for these formulas?

Answer 8

You know \(\sin \theta=\frac{3}{5}\), so \(\sin^2\theta=(\sin \theta)^2=\frac{9}{25}\)

Answer 9

I don't know the symbol that you put on top of the 25.

Answer 10

I wrote 9/25 as \(\dfrac{9}{25}\)

Answer 11

oh okay. i apologize i couldn't read that.

Answer 12

No problem, I'll try to make it larger!

Can you calculate with this and the third formula the value of \(\cos^2\theta\) ?

Answer 13

Well I think so because you need \[\cos ^{2}\theta - \sin ^{2}\theta\] to get cos 2 theta?

Answer 14

No, I think that, because \(\sin^2\theta=\dfrac{9}{25}\), that \(\cos^2\theta=\dfrac{16}{25}\), because their sum equals 1.

Answer 15

This means I also have found (take the square root) \(\cos\theta=\dfrac{4}{5}\).

Answer 16

What about tan theta/2?

Answer 17

We cannot answer that yet, because we still haven't found the hald-angle values :(

Answer 18

we would use this formula \[\tan 2 \theta = 2 \tan \theta \div 1 - \tan ^{2} \theta\]

Answer 19

Remember? You were supposed to find \(\cos\dfrac{\theta}{2}\) and \(\tan\dfrac{\theta}{2}=\dfrac{\sin\dfrac{\theta}{2}}{\cos\dfrac{\theta}{2}}\).

We only have the values \(\sin\theta=\dfrac{3}{5}\) and \(\cos\theta=\dfrac{4}{5}\).

Found by using the last of the three formulas.
So now it is time to see what the first two are for!

Answer 20

Oh. Okay. So we use sin theta and cos theta to find tan theta/2?

Answer 21

You might use your formula. I'm a little surprised you do know that one, and not the ones for \(\sin2\theta\) and \(\cos2\theta\)!

Answer 22

Yeah I'm sorry for that. I just find math really complicated. So, how would I be able to break this down without stressing over it?

Answer 23

OK, if

\(\sin2\theta=2\sin\theta\cos\theta\), then
\(\sin\theta=2\sin\dfrac{\theta}{2}\cos\dfrac{\theta}{2}\).

Because the only thing the formula says is: angles on the left hand side are double the ones on the right hand side.

Answer 24

Also,
\(\cos2\theta=2\cos^2\theta-1\) means:
\(\cos\theta=2\cos^2\dfrac{\theta}{2}-1\)