1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Vlad1618 [11]
3 years ago
13

In the triangle pictured, let A, B, C be the angles at the three vertices, and let a,b,c be the sides opposite those angles. Acc

ording to the ``law of sines,'' you always have: b a = sinB sinA Suppose that a and b are pieces of metal which are hinged at C. At first the angle A is π/4 radians=45o and the angle B is π/3 radians = 60o. You then widen A to 46o, without changing the sides a and b. Our goal in this problem is to use the tangent line approximation to estimate the angle B.
Mathematics
1 answer:
Troyanec [42]3 years ago
8 0

Answer:

Step-by-step explanation:

(a)

Consider the following:

A=\frac{\pi}{4}=45°\\\\B=\frac{\pi}{3}=60°

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}}

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}]

Thus, the angle B is function of A is, B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{A}]

Now find \frac{dB}{dA}

Differentiate implicitly the function \sin{B}=\sqrt{\frac{3}{2}}\sin{A} with respect to A to get,

\cos {B}.\frac{dB}{dA}=\sqrt{\frac{3}{2}}\cos A\\\\\frac{dB}{dA}=\sqrt{\frac{3}{2}}.\frac{\cos A}{\cos B}

b)

When A=\frac{\pi}{4},B=\frac{\pi}{3}, the value of \frac{dB}{dA} is,

\frac{dB}{dA}=\sqrt{\frac{3}{2}}.\frac{\cos {\frac{\pi}{4}}}{\cos {\frac{\pi}{3}}}\\\\=\sqrt{\frac{3}{2}}.\frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}}\\\\=\sqrt{3}

c)

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

f(x)=f'(x).(x-a)+f(a)

Here the function f(A)=B=\sin^{-1}[\sqrt{\frac{3}{2}}\sin{A}]

At A=\frac{\pi}{4}

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}

And,

f'(A)=\frac{dB}{dA}=\sqrt{3} from part b

Therefore, the linear approximation at A=\frac{\pi}{4} 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}

d)

Use part (c), when A=46°, B is approximately,

B=f(46°)=\sqrt{3}[46°-\frac{\pi}{4}]+\frac{\pi}{3}\\\\=\sqrt{3}(1°)+\frac{\pi}{3}\\\\=61.732°

You might be interested in
please help i rly need it my math teacher gave me over 30 assignments and a project that are all due tomorrow today and i want t
Sedaia [141]

Answer:

x= -14

Step-by-step explanation:

-14 subtracted by 2 is -16, since its absolute it means its positive, meaning its 16 greater than 14, I may be wrong so just to let you know. Depends on how the teacher wants it to be.

8 0
3 years ago
Solve for x<br> 20+.10x=50+.04x
maksim [4K]

Answer:

20 + .06x = 50

.06x = 30

x = 50

4 0
3 years ago
Jimmy, Joe and sally are racing their cars. Jimmy and Joe have completed the same number of laps- six fewer laps than Sally. If
shusha [124]
X = the number of laps Joe has ran.
x + x + x + 6 = 222
3x = 216
x = 72
Jimmy ran 72 laps.
Joe ran 72 laps.
Sally ran 78 laps.
3 0
3 years ago
Aubrey played on the playground for 2 and a half hours how many minutes did she play
ss7ja [257]
There are 60 minutes in one hour
multiply the number of hours by the number of minutes per hour
2.5×60=150
the answer is Aubrey played on the playground for 150 minutes
8 0
3 years ago
What is the limit in miles per hour if the speed limit is 55 km/hr??
slava [35]
The answer is 34 MPH.
4 0
3 years ago
Other questions:
  • 3x^2y^4 x 2x^6y simplified
    9·1 answer
  • A triangular space between three streets has measurements shown. how much new curbing will be needed to go around the​ space? ho
    15·1 answer
  • Explain in steps please
    15·1 answer
  • A base to the power of 0 is always A 0 B 3
    8·1 answer
  • Write an equation that represents each mixed number as a sums of a whole number and unit fractions 4 1/4 and 1 5/6
    13·1 answer
  • A company just purchased a fleet of new cars for $30,500 per car. It is estimated that value of each car will depreciate by $450
    5·1 answer
  • Josh wants to wallpaper the back wall of his bedroom. The wall is in the shape of a rectangle. Its length is 17 feet and its
    11·1 answer
  • (2 3/8 + 1 3/4) + 5 7/8<br> What is the answer
    13·2 answers
  • 420/720
    13·1 answer
  • 300% of 25$ i need a real answer
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!