# How to divide sin?

by Nano-Passion
Tags: divide
 P: 1,306 1. The problem statement, all variables and given/known data Its a word problem but I will just state everything here in simple form. Given ------------ V initial = 100ft/s r=300ft r= 1/32 V^2 Sin 2theta Unknown --------- Solve for theta 3. The attempt at a solution 300=1/32 (100)^2 sin 2theta 300= 1/32 (10000) sin 2theta 300= 312.5 sin 2theta 300/312.5= sin 2theta .96 = sin 2theta I know the next step would be to divide by sin but how???
Mentor
P: 21,407
 Quote by Nano-Passion 1. The problem statement, all variables and given/known data Its a word problem but I will just state everything here in simple form. Given ------------ V initial = 100ft/s r=300ft r= 1/32 V^2 Sin 2theta Unknown --------- Solve for theta 3. The attempt at a solution 300=1/32 (100)^2 sin 2theta 300= 1/32 (10000) sin 2theta 300= 312.5 sin 2theta 300/312.5= sin 2theta .96 = sin 2theta I know the next step would be to divide by sin but how???
"sin" is not a number, and is not multiplying 2theta. Sine is a function, so your line at the end is similar to .96 = f(2x).

If you have a variable that is the argument to a function, what can you do to get at the function's argument?
 P: 1,035 You have 2θ there. That is really like saying Sin(θ+θ) isn't it?
HW Helper
Thanks
P: 26,148
How to divide sin?

(have a theta: θ and try using the X2 tag just above the Reply box )
 Quote by Nano-Passion .96 = sin 2theta I know the next step would be to divide by sin but how???
nooo … the next step is to use sine tables (or the arcsin button on your calculator)
P: 1,306
 Quote by Mark44 "sin" is not a number, and is not multiplying 2theta. Sine is a function, so your line at the end is similar to .96 = f(2x). If you have a variable that is the argument to a function, what can you do to get at the function's argument?
hmm..

f(2x)=.96
2x=.96
x=.48??? -.-grrr

I'm having a little trouble =/ I can't seem to get the inverse ..

What I did before was I asked myself:
sin of what? = .96 <-- plugged it in calculator as sin of inverse =/
2θ = 1.287
θ = .6435

But I want to figure our whats going on in between.
 Quote by tiny-tim (have a theta: θ and try using the X2 tag just above the Reply box ) nooo … the next step is to use sine tables (or the arcsin button on your calculator)
Thanks but I want to really understand the steps in between and not just get the answer through the calculator.

And is there a button here for theta?
Mentor
P: 21,407
 Quote by Nano-Passion hmm.. f(2x)=.96 2x=.96
f is a function. What happened to it? You can't just ignore it.
 Quote by Nano-Passion x=.48??? -.-grrr I'm having a little trouble =/ I can't seem to get the inverse .. Thanks but I want to really understand the steps in between and not just get the answer through the calculator. And is there a button here for theta?
 HW Helper P: 6,202 If you have sinx = A, then x = sin-1(A).
P: 1,306
 Quote by Mark44 f is a function. What happened to it? You can't just ignore it.
I know but I didn't know what to do, can you help me?

I'm used to things such as:
f(x) = 2x+4
y = 2x+4
y/2 -2 = x
f-1x = 1/2y - 2

but when it comes to

f(2x) = .96 then I'm confused on what to do.
 P: 15 Isn't sin2$$\theta$$ a trigonometric Identity that becomes 2 sin $$\theta$$ cos$$\theta$$?
 Mentor P: 21,407 In your problem, the function is the sine function, and your equation is: sin(2x) = .96 (I'm using x instead of theta.) The thing to do is to apply the inverse sin function to both sides. sin-1(sin(2x)) = sin-1(.96) ==> 2x = sin-1(.96) ==> x = (1/2)sin-1(.96) This will give you one value for x, but it might be that your problem calls for other solutions. If so, you will need to use some of the ideas from trig to get the other solutions.
Mentor
P: 21,407
 Quote by AwesomeSN Isn't sin2$$\theta$$ a trigonometric Identity that becomes 2 sin $$\theta$$ cos$$\theta$$?
But in this problem, that's the wrong way to go. Making this replacement turns the problem into $2 sin \theta cos\theta = .96$

and this doesn't get you closer to a solution. Using inverse functions does.
P: 1,306
 Quote by Mark44 In your problem, the function is the sine function, and your equation is: sin(2x) = .96 (I'm using x instead of theta.) The thing to do is to apply the inverse sin function to both sides. sin-1(sin(2x)) = sin-1(.96) ==> 2x = sin-1(.96) ==> x = (1/2)sin-1(.96) This will give you one value for x, but it might be that your problem calls for other solutions. If so, you will need to use some of the ideas from trig to get the other solutions.
Thanks, but please bare with me, my goal is to understand everything in mathematics as much as I can. I hope I am not being a bother but I am not satisfied with knowing that I should simply multiply by the inverse on both sides.

My question is, how does sin-1 cancel with sin when you multiply them together? It doesn't sound mathematically logical? I need the steps in between.
Mentor
P: 21,407
 Quote by Nano-Passion Thanks, but please bare with me, my goal is to understand everything in mathematics as much as I can. I hope I am not being a bother but I am not satisfied with knowing that I should simply multiply by the inverse on both sides.
No, you're not being a bother. If you don't understand something it's better to keep asking questions until things are clear.

One thing that you need to understand here is that the operation is NOT multiplication. This is something you have not been clear on since your first post in this thread.

sin(2x) is sometimes written as sin 2x. In either form it is NOT sin times 2x - it's sin of 2x. Similar to what I wrote earlier - f(2x) is not f times 2x. It's f OF 2x, where f was the name of some unspecified function.
 Quote by Nano-Passion My question is, how does sin-1 cancel with sin when you multiply them together?
They're not being multiplied. What is happening is that I am forming a composite function. Whenever you have a function that has an inverse, applying them together in either order gives you the identity function, the function that leaves its argument completely unchanged.

For example, if f(x) = 2x + 3, then the inverse of this function is f-1(x) = (x - 3)/2.

f(f-1(2) = 2 and f-1(f(0)) = 0.
You can verify these statements by using the formulas for the functions.

In a similar way sin(sin-1(x) = x, but there are some restrictions of the values of x that are allowed. Also, in the opposite order, sin-1(sin(x)) = x, and there are some restrictions here, as well. In the first equation, x has to be between -1 and +1, inclusive. In the second equation, x has to be between -pi/2 and +pi/2 if you're working in radians, or between -90 deg and +90 deg, if you're working in degrees.
 Quote by Nano-Passion It doesn't sound mathematically logical? I need the steps in between.
 P: 163 Yes, sin-1(sin(theta) ) = theta
HW Helper
P: 1,347
 Quote by RocketSci5KN Yes, sin-1(sin(theta) ) = theta
That's not always true. Did you not read Mark44's post above?

For instance, what is
$$\sin^{-1} (\sin 3\pi/4)$$
? Hint: it's not 3π/4.
Emeritus