11) Using Eulerβs Formula (πππ = Cos π + π
The exact value of cos(45) is β22 / 2 or approximately 0.7071. This can be found by using trigonometric methods to evaluate cos(225Β°) as -β22 / 2. In a video tutorial, it is demonstrated that using a calculator will result in an approximate value of -0.7071 . The exact value as seen in a table is -β(2) / 2.
For the complex numbers in exponential form ( r e^{i\theta} ), we can convert them to trigonometric form ( r(\cos \theta + i \sin \theta) ) and then standard form ( a + bi ). Here are the conversions for the given complex numbers:
a. For ( 3e^{225^\circ i} ): First, let's express it in trigonometric form: ( 3(\cos 225^\circ + i \sin 225^\circ) ) After calculating (\cos 225^\circ) and (\sin 225^\circ), we can represent it in standard form ( a + bi ).
b. For ( 2e^{\frac{11\pi}{6}i} ): Similarly, first, let's express it in trigonometric form: ( 2(\cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6}) ) After calculating (\cos \frac{11\pi}{6}) and (\sin \frac{11\pi}{6}), we can represent it in standard form ( a + bi ).
c. For ( 5e^{\frac{3\pi}{2}i} ): Once more, let's express it in trigonometric form: ( 5(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) ) After calculating (\cos \frac{3\pi}{2}) and (\sin \frac{3\pi}{2}), we can represent it in standard form ( a + bi ).
Shall we proceed with the calculations to convert these complex numbers into trigonometric and standard forms?
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