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Simplify (cos(425)-tanh (0.3+j0.7) cosh (In (3+5)3-15) and write your final answer in polar form.

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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:

User Pinakin Shah
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