Solving the Equation sinx 2 in the Complex Plane
When tackling the equation sinx 2, it's crucial to understand the fundamental properties and constraints of the sine function. The sine of any angle x is defined within a specific range, and the primary function of sine only outputs values between -1 and 1. This means that the equation sinx 2 has no real solutions because the sine function, in the context of real numbers, is confined to the interval [-1, 1].
Exploring the Complex Solutions
However, by delving into the complex number system, we can explore the possibility of solutions. To do this, we will rewrite the sine formula in terms of complex exponentials. The sine of a complex number x can be expressed as:
[ sin x frac{e^{frac{pi i x}{180}} - e^{-frac{pi i x}{180}}}{2i} ]
Substituting x piq, we get:
[ frac{e^{frac{pi i q}{180}} - e^{-frac{pi i q}{180}}}{2i} 2 ]
To eliminate the imaginary part, we need to set the cosine part equal to zero:
[ cos q 0 ]
The solutions for cos q 0 are given by:
[ q 90, 270, 450, ... ]
or in general:
[ q (2k 1)90°, quad k in mathbb{Z} ]
Substituting ( q 90 ) and ( q -90 ) back into the equation, we get:
[ frac{e^{frac{90 pi i}{180}} - e^{-frac{90 pi i}{180}}}{2i} 2 ]
and
[ frac{e^{-frac{90 pi i}{180}} - e^{frac{90 pi i}{180}}}{2i} 2 ]
Let's simplify these expressions:
[ frac{e^{frac{pi i}{2}} - e^{-frac{pi i}{2}}}{2i} 2 ]
and
[ frac{e^{-frac{pi i}{2}} - e^{frac{pi i}{2}}}{2i} 2 ]
Since ( e^{frac{pi i}{2}} i ) and ( e^{-frac{pi i}{2}} -i ), we get:
[ frac{i i}{2i} 2 quad text{or} quad frac{-i - i}{2i} 2 ]
Simplifying further:
[ frac{2i}{2i} 1 quad text{or} quad frac{-2i}{2i} -1 ]
Add appropriate factors:
[ i^2 -1 ]
and
[ -i^2 1 ]
Next, we solve for the real part to find q:
[ frac{e^{frac{pi q}{180}} e^{-frac{pi q}{180}}}{2i} 2 ]
Combining the exponents:
[ e^{frac{pi q}{180} - frac{pi q}{180}} e^0 1 ]
Thus:
[ 1 - 2i 4 ]
From here, we take the logarithm of both sides:
[ frac{pi q}{180} ln(2 sqrt{3}) quad text{or} quad frac{pi q}{180} ln(2 - sqrt{3}) ]
Solving for q:
[ q frac{180 ln(2 sqrt{3})}{pi} approx 75.4561 ]
and
[ q frac{180 ln(2 - sqrt{3})}{pi} approx -75.4561 ]
Therefore, the solutions for x are:
[ x 90k frac{180 ln(2 sqrt{3})}{pi} ]
or
[ x 90k - frac{180 ln(2 - sqrt{3})}{pi} ]
where k is any integer.
Conclusion
In summary, while the equation sinx 2 has no real solutions due to the limitations of the sine function, it can be solved in the complex plane. By applying complex numbers and logarithmic transformations, we were able to find the complex solutions to this trigonometric equation. These solutions provide a deeper understanding of the behavior of sine in the broader context of complex analysis.