Trig Equations: arcsin x = arccos(3/5) - arcsin(4/5), etc

axrw

New member
Joined
Mar 18, 2007
Messages
44
I've got two problems that I'm not sure what to do on.

arcsin x = arccos ( 3/5) - arcsin ( 4/5)

and

Suppose that sin x = 5 cos x. Find sin x * cos x.

On the first I thought it would be something like:

x = cos( arccos(3/5) - arcsin( 4/5))

But I don't know what to do with that.

Thanks for any help.

EDIT: Ok, I just figured out the first one. I forgot about cosine sum identity. I still can't get the second though.
 
Re: Trig Equations.

axrw said:
I've got two problems that I'm not sure what to do on.

arcsin x = arccos ( 3/5) - arcsin ( 4/5)

and

Suppose that sin x = 5 cos x. Find sin x * cos x.

tan x = 5

sec^2(x) = 26

cos^2(x) = 1/26

Now continue......


On the first I thought it would be something like:

x = cos( arccos(3/5) - arcsin( 4/5))

But I don't know what to do with that.

Thanks for any help.

EDIT: Ok, I just figured out the first one. I forgot about cosine sum identity. I still can't get the second though.
 
Thank you! Another identity I missed, I see. These identities will be the death of me. :(
 
Re: Trig Equations: arcsin x = arccos(3/5) - arcsin(4/5), et

Hello, axrw!

I caught an "eyeball" solution for the first one . . .


arcsinx=arccos(35)arcsin(45)\displaystyle \arcsin x \:= \:\underbrace{\arccos\left(\frac{3}{5}\right)}\, -\,\arcsin\left(\frac{4}{5}\right)

Let arccos(35)=θ        cosθ=35=adjhyp\displaystyle \arccos\left(\frac{3}{5}\right) \,=\,\theta\;\;\Rightarrow\;\;\cos\theta \,=\,\frac{3}{5} \,=\,\frac{adj}{hyp}

θ\displaystyle \theta is in a right triangle with: adj=3,  hyp=5\displaystyle \,adj \,=\,3,\;hyp\,=\,5
From Pythagorus, we find that: opp=4        sinθ=45\displaystyle \,opp\,=\,4\;\;\Rightarrow\;\;\sin\theta\,=\,\frac{4}{5}
. . Hence: θ=arcsin(45)\displaystyle \,\theta \,=\,\arcsin\left(\frac{4}{5}\right)

The equation becomes: arcsinx  =  arcsin(45)arcsin(45)  =  0\displaystyle \:\arcsin x \;=\;\arcsin\left(\frac{4}{5}\right)\,-\,\arcsin\left(\frac{4}{5}\right) \;=\;0

Therefore: x=0\displaystyle \:x \:=\:0

 
Thank you soroban. Your way is a bit more efficient than what I did.
 
Top