Answer:
t = 1
Explanation:
Thinking process:
Let the value of the expression be:
[tex]y = (t + \frac{\pi }{4}) + sint + \frac{1}{8}sin 3t[/tex]
using the Fourier analysis and approximations gives
t = 1
The value of [tex]t[/tex] is near to [tex]0.0399[/tex].
Fourier series is a very powerful tool in connection with various problems involving partial differential equations. Their representation in terms of simple periodic functions such as sine and cosine function, which leads to Fourier series (FS).
Fourier analysis of the instantaneous value of a waveform can be represented by,
[tex]y=t+\frac{\pi }{4}+sint+\frac{1}{8}sin3t[/tex]
Apply the Newton Raphson method to determine the value of [tex]t[/tex] when [tex]y=0.88[/tex] using [tex]r_{1} =0.04[/tex]. Correct to 4 decimal places.
Newton-Raphson Method:
If [tex]x_{a}[/tex] is an approximation a solution of [tex]f(x)=0[/tex] and [tex]{f}'\left ( x_{a} \right )\neq 0[/tex] the next approximation is given by,
[tex]x_{n-1}=x_{a}-\frac{f\left ( x_{a} \right )}{{f}'\left ( x_{a} \right )} \\ y=t+\frac{\pi }{4}+sint+\frac{1}{8}sin3t[/tex]
[tex]t+\frac{\pi }{4}+sint+\frac{1}{8}sin3t=0.88\Rightarrow f(t)=sin \ t+\frac{1}{8}sin3t+t+\frac{\pi }{4}-0.88[/tex]
[tex]{f}'\left ( t \right )=cost+\frac{3}{8}cos3t+1 \\ t_{1}=0.04[/tex]
[tex]=t_{n}-\frac{sint_{n}+\frac{1}{8}sin3t_{n}+t_{n}+\frac{\pi }{4}-0.88}{cost_{n}+\frac{3}{8}cos3t_{n}+1}[/tex]
Below attaching the answer calculation.
Learn more about the topic Fourier Series or analysis:
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