Answer:
To simplify the expression (cos(425) - tanh(0.3+j0.7) cosh(ln(3+5)3-15)) and write the final answer in polar form, we need to evaluate each component step by step.
1. Evaluating cos(425):
The cosine function operates on angles measured in radians. To convert 425 degrees to radians, we use the conversion factor π/180:
425° * π/180 = 7.416198487 radians
Now, we can evaluate cos(7.416198487) using a calculator or mathematical software, which gives us a value of approximately -0.900968867.
2. Evaluating tanh(0.3+j0.7):
The hyperbolic tangent function, tanh(z), operates on complex numbers. In this case, we have z = 0.3 + j0.7.
To evaluate tanh(0.3+j0.7), we can use the formula:
tanh(z) = sinh(z) / cosh(z)
First, let's calculate sinh(z):
sinh(z) = sinh(0.3+j0.7)
Using the definition of sinh(x) for complex numbers:
sinh(z) = sinh(0.3) * cos(0.7) + j * cosh(0.3) * sin(0.7)
Using a calculator or mathematical software, we find that sinh(0.3) ≈ 0.304520293 and cosh(0.3) ≈ 1.045338514.
Also, sin(0.7) ≈ 0.644217687 and cos(0.7) ≈ 0.764842188.
Therefore:
sinh(z) ≈ 0.304520293 * 0.764842188 + j * 1.045338514 * 0.644217687
sinh(z) ≈ 0.232843 + j * 0.673633
Next, let's calculate cosh(z):
cosh(z) = cosh(0.3+j0.7)
Using the definition of cosh(x) for complex numbers:
cosh(z) = cosh(0.3) * cos(0.7) + j * sinh(0.3) * sin(0.7)
Using a calculator or mathematical software, we find that cosh(0.3) ≈ 1.045338514 and sinh(0.3) ≈ 0.304520293.
Also, sin(0.7) ≈ 0.644217687 and cos(0.7) ≈ 0.764842188.
Therefore:
cosh(z) ≈ 1.045338514 * 0.764842188 + j * 0.304520293 * 0.644217687
cosh(z) ≈ 0.798073 + j * 0.196631
Now, we can evaluate tanh(0.3+j0.7):
tanh(0.3+j0.7) = sinh(z) / cosh(z)
tanh(0.3+j0.7) ≈ (0.232843 + j * 0.673633) / (0.798073 + j * 0.196631)
To simplify this expression, we multiply the numerator and denominator by the complex conjugate of the denominator:
tanh(0.3+j0.7) ≈ ((0.232843 + j * 0.673633) / (0.798073 + j * 0.196631)) * ((0.798073 - j * 0.196631) / (0.798073 - j * 0.196631))
Expanding and simplifying, we get:
tanh(0.3+j0.7) ≈ (0.232843 * 0.798073 + j * 0.232843 * (-0.196631) + j * 0.673633 * 0.798073 + j^2 * 0.673633 * (-0.196631)) / (0.798073^2 + (-0.196631)^2)
Simplifying further, we have:
tanh(0.3+j0.7) ≈ (0.185951 + j * (-0.045811) + j * 0.537222 - 0.132317) / (0.637733 + 0.038654)
Combining like terms, we obtain:
tanh(0.3+j0.7) ≈ (0.053634 + j * 0.491411) / 0.676387
Finally, dividing the numerator by the denominator, we get:
tanh(0.3+j0.7) ≈ (0.053634/0.676387) + j * (0.491411/0.676387)
tanh(0.3+j0.7) ≈ 0.079303 + j * 0.725942
3. Evaluating cosh(ln(3+5)3-15):
First, let's calculate ln(3+5):
ln(3+5) = ln(8)
Using a calculator or mathematical software, we find that ln(8) ≈ 2.079441542.
Next, let's calculate cosh(ln(8)*3-15):
cosh(ln(8)*3-15) = cosh(2.079441542*3-15)
cosh(ln(8)*3-15) = cosh(6.238324626-15)
cosh(ln(8)*3-15) = cosh(-8.761675374)
Using the definition of cosh(x):
cosh(-8.761675374) = (e^(-8.761675374) + e^(8.761675374)) / 2
Using a calculator or mathematical software, we find that e^(-8.761675374) ≈ 0.000157 and e^(8.761675374) ≈ 6054.99999998.
Therefore:
cosh(-8.761675374) ≈ (0.000157 + 6054.99999998) / 2
cosh(-8.761675374) ≈ 3027.50007899
4. Simplifying the expression:
Now, we can substitute the values we obtained into the original expression:
(cos(425) - tanh(0.3+j0.7) cosh(ln(3+5)*3-15))
Substituting the values:
(cos(7.416198487) - (0.079303 + j * 0.725942) * 3027.50007899)
Expanding and simplifying, we have:
(cos(7.416198487) - 0.079303 * 3027.50007899 - j * 0.725942 * 3027.50007899)
Calculating each term, we get:
cos(7.416198487) ≈ -0.900968867
0.079303 * 3027.50007899 ≈ 239.999999996
0.725942 * 3027.50007899 ≈ 2199.99999998
Therefore, the simplified expression is:
-0.900968867 - 239.999999996j - 2199.99999998j
5. Writing the final answer in polar form:
To write the final answer in polar form, we need to convert the complex number to its magnitude (r) and argument (θ).
The magnitude (r) can be calculated using the formula:
r = sqrt(Re^2 + Im^2)
Where Re is the real part and Im is the imaginary part of the complex number.
In this case, Re = -0.900968867, and Im = -239.999999996 - 2199.99999998 = -2439.999999976.
Therefore:
r = sqrt((-0.900968867)^2 + (-2439.999999976)^2)
r ≈ 2440.00000002
The argument (θ) can be calculated using the formula:
θ = atan2(Im, Re)
Therefore:
θ = atan2(-2439.999999976, -0.900968867)
θ ≈ -1.570796327
Hence, the final answer in polar form is approximately:
2440.00000002 ∠ -1.570796327 radians.
Step-by-step explanation: