$ k=2 $: $ e^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} $
["Understanding the Complex Exponential: ( e^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} )", "In the fascinating world of complex numbers, Euler’s formula reveals deep connections between trigonometry and exponential functions. One of the most illustrative examples is ( e^{i\ heta} ), which describes points on the unit circle in the complex plane. A key familiar angle is ( \ heta = \frac{5\pi}{4} ), and exploring ( e^{i5\pi/4} ) helps uncover how complex exponentials translate into real and imaginary components.", "What Does ( e^{i5\pi/4} ) Represent?", "According to Euler’s formula:", "[\ne^{i\ heta} = \cos\ heta + i\sin\ heta\n]", "Substituting ( \ heta = \frac{5\pi}{4} ), we compute:", "[\ne^{i5\pi/4} = \cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\n]", "The angle ( \frac{5\pi}{4} ) radians is equivalent to ( 225^\circ ), located in the third quadrant of the unit circle, where both cosine and sine are negative.", "Trigonometric Values at ( \frac{5\pi}{4} )", "At this angle, reference angle is ( \frac{\pi}{4} ) (45°), and both sine and cosine have magnitude ( \frac{\sqrt{2}}{2} ). In the third quadrant, signs are negative:", "[\n\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\n]", "Substituting these values gives:", "[\ne^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\n]", "This elegant result confirms the geometric interpretation: the complex number lies exactly on the unit circle at ( 225^\circ ), with balanced negative real and imaginary parts.", "Significance in Complex Analysis", "Understanding ( e^{i5\pi/4} ) illuminates foundational concepts in complex analysis, signal processing, and quantum mechanics, where complex exponentials model periodic behavior, rotations, and wave functions. The explicit form ( -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} ) serves as a precise coordinate, bridging algebra and geometry.", "Conclusion", "The identity ( e^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} ) exemplifies how complex exponentials simplify trigonometric expressions. Recognizing such equivalences empowers deeper insight into advanced mathematical frameworks and real-world applications involving oscillations and rotations.", "---", "Keywords: ( e^{i5\pi/4} ), complex exponential, Euler’s formula, ( \frac{5\pi}{4} ), complex numbers, unit circle, Trigonometry, complex analysis, radians, imaginary part, real part, ( -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} ), mathematical identity, trigonometric values.", "Meta Description:\nDiscover why ( e^{i5\pi/4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} ) reveals the link between complex exponentials and trigonometry, explaining the calculation and significance in math and science."]
