I believe if that says three minus four times three minus five or three plus negative four times three plus negative five (same thing) then the answers should be positive two (2.)
C. Revising because it should be your last stage in the writing process before you publish
140 1/8 can not be expressed in radical form because it neither has an exponent or a square root. Unless if 140 is raised to the power of 1/8, then that can be.
(3x) + (x+60) = 180
4x + 60 = 180
Subtract 60 from both sides
4x = 120
Divide 4 from both sides
x = 30
You want to end up with
. Expand this using the angle sum identity for sine:
![A\sin(\omega t+\phi)=A\sin(\omega t)\cos\phi+A\cos(\omega t)\sin\phi](https://tex.z-dn.net/?f=A%5Csin%28%5Comega%20t%2B%5Cphi%29%3DA%5Csin%28%5Comega%20t%29%5Ccos%5Cphi%2BA%5Ccos%28%5Comega%20t%29%5Csin%5Cphi)
We want this to line up with
. Right away, we know
.
We also need to have
![\begin{cases}A\cos\phi=2\\A\sin\phi=5\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7DA%5Ccos%5Cphi%3D2%5C%5CA%5Csin%5Cphi%3D5%5Cend%7Bcases%7D)
Recall that
for all
; this means
![(A\cos\phi)^2+(A\sin\phi)^2=2^2+5^2\implies A^2=29\implies A=\sqrt{29}](https://tex.z-dn.net/?f=%28A%5Ccos%5Cphi%29%5E2%2B%28A%5Csin%5Cphi%29%5E2%3D2%5E2%2B5%5E2%5Cimplies%20A%5E2%3D29%5Cimplies%20A%3D%5Csqrt%7B29%7D)
Then
![\begin{cases}\cos\phi=\frac2{\sqrt{29}}\\\sin\phi=\frac5{\sqrt{29}}\end{cases}\implies\tan\phi=\dfrac{\sin\phi}{\cos\phi}=\dfrac52\implies\phi=\tan^{-1}\left(\dfrac52\right)](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%5Ccos%5Cphi%3D%5Cfrac2%7B%5Csqrt%7B29%7D%7D%5C%5C%5Csin%5Cphi%3D%5Cfrac5%7B%5Csqrt%7B29%7D%7D%5Cend%7Bcases%7D%5Cimplies%5Ctan%5Cphi%3D%5Cdfrac%7B%5Csin%5Cphi%7D%7B%5Ccos%5Cphi%7D%3D%5Cdfrac52%5Cimplies%5Cphi%3D%5Ctan%5E%7B-1%7D%5Cleft%28%5Cdfrac52%5Cright%29)
So we end up with
![2\sin(4\pi t)+5\cos(4\pi t)=\sqrt{29}\sin\left(4\pi t+\tan^{-1}\left(\dfrac52\right)\right)](https://tex.z-dn.net/?f=2%5Csin%284%5Cpi%20t%29%2B5%5Ccos%284%5Cpi%20t%29%3D%5Csqrt%7B29%7D%5Csin%5Cleft%284%5Cpi%20t%2B%5Ctan%5E%7B-1%7D%5Cleft%28%5Cdfrac52%5Cright%29%5Cright%29)