Answer:
Step-by-step explanation:
(a)
Consider the following:

Use sine rule,
![\frac{b}{a}=\frac{\sinB}{\sin A} \\\\=\frac{\sin{\frac{\pi}{3}} }{\sin{\frac{\pi}{4}}}\\\\=\frac{[\frac{\sqrt{3}}{2}]}{\frac{1}{\sqrt{2}}}\\\\=\frac{\sqrt{2}}{2}\times \frac{\sqrt{2}}{1}=\sqrt{\frac{3}{2}}](https://tex.z-dn.net/?f=%5Cfrac%7Bb%7D%7Ba%7D%3D%5Cfrac%7B%5CsinB%7D%7B%5Csin%20A%7D%0A%5C%5C%5C%5C%3D%5Cfrac%7B%5Csin%7B%5Cfrac%7B%5Cpi%7D%7B3%7D%7D%0A%7D%7B%5Csin%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B%5B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%5D%7D%7B%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%7D%5C%5C%5C%5C%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%5Ctimes%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B1%7D%3D%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D)
Again consider,
![\frac{b}{a}=\frac{\sin{B}}{\sin{A}} \\\\\sin{B}=\frac{b}{a}\times \sin{A}\\\\\sin{B}=\sqrt{\frac{3}{2}}\sin {A}\\\\B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{A}]](https://tex.z-dn.net/?f=%5Cfrac%7Bb%7D%7Ba%7D%3D%5Cfrac%7B%5Csin%7BB%7D%7D%7B%5Csin%7BA%7D%7D%0A%5C%5C%5C%5C%5Csin%7BB%7D%3D%5Cfrac%7Bb%7D%7Ba%7D%5Ctimes%20%5Csin%7BA%7D%5C%5C%5C%5C%5Csin%7BB%7D%3D%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Csin%20%7BA%7D%5C%5C%5C%5CB%3D%5Csin%5E%7B-1%7D%5B%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Csin%7BA%7D%5D)
Thus, the angle B is function of A is, ![B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{A}]](https://tex.z-dn.net/?f=B%3D%5Csin%5E%7B-1%7D%5B%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Csin%7BA%7D%5D)
Now find 
Differentiate implicitly the function
with respect to A to get,

b)
When
, the value of
is,

c)
In general, the linear approximation at x= a is,

Here the function ![f(A)=B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{A}]](https://tex.z-dn.net/?f=f%28A%29%3DB%3D%5Csin%5E%7B-1%7D%5B%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Csin%7BA%7D%5D)
At 
![f(\frac{\pi}{4})=B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{\frac{\pi}{4}}]\\\\=\sin^{-1}[\sqrt{\frac{3}{2}}.\frac{1}{\sqrt{2}}]\\\\\=\sin^{-1}(\frac{\sqrt{2}}{2})\\\\=\frac{\pi}{3}](https://tex.z-dn.net/?f=f%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29%3DB%3D%5Csin%5E%7B-1%7D%5B%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Csin%7B%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%5D%5C%5C%5C%5C%3D%5Csin%5E%7B-1%7D%5B%5Csqrt%7B%5Cfrac%7B3%7D%7B2%7D%7D.%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5D%5C%5C%5C%5C%5C%3D%5Csin%5E%7B-1%7D%28%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%5C%5C%5C%5C%3D%5Cfrac%7B%5Cpi%7D%7B3%7D)
And,
from part b
Therefore, the linear approximation at
is,
![f(x)=f'(A).(x-A)+f(A)\\\\=f'(\frac{\pi}{4}).(x-\frac{\pi}{4})+f(\frac{\pi}{4})\\\\=\sqrt{3}.[x-\frac{\pi}{4}]+\frac{\pi}{3}](https://tex.z-dn.net/?f=f%28x%29%3Df%27%28A%29.%28x-A%29%2Bf%28A%29%5C%5C%5C%5C%3Df%27%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29.%28x-%5Cfrac%7B%5Cpi%7D%7B4%7D%29%2Bf%28%5Cfrac%7B%5Cpi%7D%7B4%7D%29%5C%5C%5C%5C%3D%5Csqrt%7B3%7D.%5Bx-%5Cfrac%7B%5Cpi%7D%7B4%7D%5D%2B%5Cfrac%7B%5Cpi%7D%7B3%7D)
d)
Use part (c), when
, B is approximately,
![B=f(46°)=\sqrt{3}[46°-\frac{\pi}{4}]+\frac{\pi}{3}\\\\=\sqrt{3}(1°)+\frac{\pi}{3}\\\\=61.732°](https://tex.z-dn.net/?f=B%3Df%2846%C2%B0%29%3D%5Csqrt%7B3%7D%5B46%C2%B0-%5Cfrac%7B%5Cpi%7D%7B4%7D%5D%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%5C%5C%5C%5C%3D%5Csqrt%7B3%7D%281%C2%B0%29%2B%5Cfrac%7B%5Cpi%7D%7B3%7D%5C%5C%5C%5C%3D61.732%C2%B0)
True because they have the same size angles
Answer:
The average monthly expenditure is 1,155.35 $.
Step-by-step explanation:
The average of any sequency is given by the sum of it's individual parts divided by the number of parts it has. So in this case the average of monthly expenditure will be the sum of all individual expenditures divided by the number of months. The question can be solved like this:
average = (March+ April + May + June + July+ August)/6
average = (1,249.59 + 1,365.38 + 1,024.3 + 1,100.4 + 992,4 + 1,200.02)/6
average = 1,155.35 $
Answer:
x+2
Step-by-step explanation:
we know that
To find out the factors of a polynomial, determine the x-intercepts or zeros of the polynomial
Remember that
The x-intercepts of a polynomial are the values of x when the value of the polynomial is equal to zero
In this problem
Looking at the graph
For x=-2, f(x)=0
so
x=-2 is an x-intercept or zero of the polynomial
To find out the factor move the constant to the left side and equate to zero

adds 2 both sides

therefore
The factor is (x+2)
D/dx ( cos (4x)) = - sin (4x) · d/dx ( 4x) = - 4 sin (4x)