Possible complex angles with no imaginary periodicity

In summary, the conversation discusses the results of plugging in a larger imaginary number for the known equation sin^2(x) + cos^2(x) = 1, and the discovery of periodicity and potential implications for a different approach to trigonometry. It also introduces the concept of imaginary angles and their relationship to hyperbolic functions.
  • #1
Some_dude91
4
0
Trying the already known equation, sin^2(x) + cos^2(x) = 1 i wondered what would happen
if i took that either sin(x) or cos(x) squared equalled a number greater than 1, so when i plugged in sin(x) as 5/3 i got cos(x) 4i/3 ,
went to euler's equation and added the results,
then put the aggregate to the logarithm and got
angle = 2kp + pi/2 ± 1.0986i. Solving for cos(x) as well i got an imaginary angle equal to
angle = ±1.0986i,
but as i went on having the number grow larger, the imaginary part would grow proportionately. Now, based on the results i got, i saw such angles only poccessed periodicity on the real, unchanged, part whereas the imaginary ones poccessed none. The plus or minus part was due to the fact that for 1 possible cosine (or sine), the resulting sine (or cosine) was either positive or negative, on the equation
sin^2(x) + cos^2(x) = 1

I don't know if this is wrong but it yields an interesting result that might be telling something about a different, outside the usual trigonometry approach. However, seeing as i treated cosine and sine as continuous functions, and knowing that they can be used only to satisfy an equation, just numerically, i might be wrong, but based on the modification on the exponential function by euler,
e^x = cosh(x) + sinh(x) converted to e^ix = cos(x) + isin(x) i did the inverse, and got that for imaginary angles, cos(x) becomes cosh(x) and sin(x) becomes i*sinh(x), based on an approach
by the taylor series.

Also by using the trigonometrical identities sin(x+y) and cos(x+y) i got that
sin(x+iy) = sin(x)cosh(y) + i*cos(x)sinh(y) and
cos(x+iy) = cos(x)cosh(x) - i*sin(x)sinh(y) which in fact do satisfy the equation i did.
Could there really be such angles? Could this approach be right?
 
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  • #2
The term angle usually means a real number. However sin(z) or cos(z) where z is a complex number is a well defined function. When z is purely imaginary, then hyperbolic functions will be equivalent.

cos(ix) = cosh(x), sin(ix) = isinh(x)
 

Related to Possible complex angles with no imaginary periodicity

1. What are complex angles?

Complex angles are angles that involve both a real and an imaginary component. They can be represented in the form a + bi, where a is the real part and bi is the imaginary part.

2. What is periodicity in relation to complex angles?

Periodicity refers to the property of a function or curve to repeat itself at regular intervals. In the case of complex angles, periodicity refers to the pattern of the imaginary component repeating itself, forming a periodic curve.

3. Can complex angles have no imaginary periodicity?

Yes, it is possible for complex angles to have no imaginary periodicity. This means that the imaginary component does not repeat itself in a regular pattern, making the angle more complex to analyze.

4. How can complex angles with no imaginary periodicity be represented?

Complex angles with no imaginary periodicity can be represented in the form a + bi, where a is the real part and bi is the non-repeating imaginary part. They can also be graphed on a complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.

5. What are the applications of studying complex angles with no imaginary periodicity?

Studying complex angles with no imaginary periodicity can be useful in various fields such as engineering, physics, and mathematics. It can help solve problems involving oscillatory behavior, stability analysis, and signal processing.

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