Emergency A Half Angle Question.

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In summary, to verify the trigonometric equation tanh(z/2) = [sinh x + i sin y] / [cosh x + cos y], where z is a complex number, one can use the identities for sine, cosine, tangent, and cotangent in complex numbers. Alternatively, one can simplify the equation to tanh(z/2) = e^z-1 / e^z+1 and solve from the right hand side.
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Homework Statement


How to verify this trigonometric equation? tanh z/2?
verify that:

tanh (z/2) = [sinh x + i sin y] / [cosh x + cos y]


z is a complex number.

How can i verify this equation?


Homework Equations


9957458708fa15c0a09a44ef7e3fbef8.png


http://mathworld.wolfram.com/images/equations/HyperbolicTangent/Inline3.gif
# sin(x + i y) = sin(x) cosh(y) + i cos(x) sinh(y).
# cos(x + i y) = cos(x) cosh(y) - i sin(x) sinh(y).
# tan(x + i y) = (tan(x) + i tanh(y)) / (1 - i tan(x) tanh(y)).
# cot(x + i y) = (cot(x) coth(y) - i) / (i cot(x) + coth(y)).




The Attempt at a Solution



i have tried to solve from right hand side but it didn't work, then i have simplified the equation tanh(z/2) = e^z-1 / e^z+1


if you help me, i will be appreciate.
thanx a lot.
 
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1. What is an emergency half angle question?

An emergency half angle question is a mathematical problem that involves finding the value of a half-angle, given the value of its corresponding full angle. This type of question is commonly encountered in trigonometry and geometry problems.

2. How do I solve an emergency half angle question?

To solve an emergency half angle question, you can use the half angle identity formula, which states that the value of a half-angle is equal to the square root of [(1-cosθ)/2], where θ is the value of the corresponding full angle. Alternatively, you can use a calculator or online calculator tool to find the numerical value of the half-angle.

3. What is the importance of understanding emergency half angle questions?

Understanding emergency half angle questions is important because it allows you to solve a wide range of mathematical problems involving angles and triangles. It is also a fundamental concept in trigonometry and can help in real-world applications such as navigation, engineering, and surveying.

4. Are there any tips for solving emergency half angle questions?

One helpful tip for solving emergency half angle questions is to always check if the given angle is in radians or degrees. If it is in radians, you will need to convert it to degrees before using the half angle identity formula. It is also important to check your final answer to ensure it makes sense in the context of the problem.

5. Can I use the half angle formula for any angle?

No, the half angle formula can only be used for acute angles (less than 90 degrees) and certain special angles such as 30, 45, and 60 degrees. For obtuse angles (greater than 90 degrees), you will need to use other trigonometric identities and formulas to find the value of the half-angle.

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