To find the rectangular form of the complex number [tex]\( 17\left(\cos(315^\circ) + i \sin(315^\circ)\right) \)[/tex], we need to evaluate the real and imaginary parts of the expression.
1. Start by recalling the form of a complex number in polar coordinates, which is given by:
[tex]\[
r(\cos(\theta) + i \sin(\theta))
\][/tex]
where [tex]\( r \)[/tex] is the modulus and [tex]\( \theta \)[/tex] is the argument (angle) of the complex number.
2. In this particular problem, the modulus [tex]\( r \)[/tex] is 17 and the argument [tex]\( \theta \)[/tex] is 315 degrees.
3. Convert 315 degrees to radians as follows:
[tex]\[
\theta = 315^\circ \times \frac{\pi}{180^\circ} = \frac{315\pi}{180} = \frac{7\pi}{4} \text{ radians}
\][/tex]
4. Now, compute the cosine and sine of [tex]\( \frac{7\pi}{4} \)[/tex]:
- [tex]\(\cos\left(\frac{7\pi}{4}\right) = \cos(315^\circ)\)[/tex]
- [tex]\(\sin\left(\frac{7\pi}{4}\right) = \sin(315^\circ)\)[/tex]
By looking at the unit circle:
- [tex]\(\cos(315^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin(315^\circ) = -\frac{\sqrt{2}}{2}\)[/tex]
5. Substitute these values into the rectangular form of the complex number:
[tex]\[
17 \left(\cos(315^\circ) + i \sin(315^\circ)\right) = 17 \left(\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right)\right)
\][/tex]
6. Apply the modulus 17:
[tex]\[
17 \left(\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right)\right) = 17 \cdot \frac{\sqrt{2}}{2} + i \cdot 17 \left(-\frac{\sqrt{2}}{2}\right)
\][/tex]
[tex]\[
= \frac{17 \sqrt{2}}{2} - i \frac{17 \sqrt{2}}{2}
\][/tex]
7. Separate the real and imaginary parts:
[tex]\[
= \frac{17 \sqrt{2}}{2} - i \frac{17 \sqrt{2}}{2}
\][/tex]
Therefore, the rectangular form of the complex number is:
[tex]\[
\boxed{\frac{17 \sqrt{2}}{2} - \frac{17 \sqrt{2}}{2}i}
\][/tex]
Given the multiple-choice options, this corresponds to:
[tex]\[
\frac{17 \sqrt{2}}{2} - \frac{17 \sqrt{2}}{2}
\][/tex]
