To find the trigonometric form of the complex number [tex]\( z = i^{22} + i^{24n - 13} \)[/tex], we need to follow these steps:
1. Evaluate [tex]\( i^{22} \)[/tex]:
The imaginary unit [tex]\( i \)[/tex] is defined as [tex]\( i = \sqrt{-1} \)[/tex]. The powers of [tex]\( i \)[/tex] cycle every four terms:
[tex]\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\][/tex]
To find [tex]\( i^{22} \)[/tex], we find the remainder when 22 is divided by 4:
[tex]\[
22 \div 4 = 5 \text{ remainder } 2
\][/tex]
Hence,
[tex]\[
i^{22} = i^2 = -1
\][/tex]
2. Evaluate [tex]\( i^{24n - 13} \)[/tex]:
Similarly, we need to express [tex]\( 24n - 13 \)[/tex] in terms of the remainder when divided by 4.
[tex]\[
24n - 13 \div 4 = 6n - 3 \text{ remainder } (24n - 13) \pmod{4}
\][/tex]
Since [tex]\( 24n \)[/tex] is always a multiple of 4,
[tex]\[
24n - 13 \equiv -13 \pmod{4} \equiv 3 \pmod{4}
\][/tex]
Therefore,
[tex]\[
i^{24n - 13} = i^3 = -i
\][/tex]
3. Combine the results to form [tex]\( z \)[/tex]:
[tex]\[
z = i^{22} + i^{24n - 13} = -1 + (-i) = -1 - i
\][/tex]
4. Convert to trigonometric form:
A complex number [tex]\( z = a + bi \)[/tex] can be expressed in polar (trigonometric) form as [tex]\( z = re^{i\theta} \)[/tex], where:
- [tex]\( r \)[/tex] is the magnitude (or modulus) of the complex number.
- [tex]\( \theta \)[/tex] is the argument (or angle) of the complex number.
For [tex]\( z = -1 - i \)[/tex]:
[tex]\[
r = \sqrt{(-1)^2 + (-i)^2} = \sqrt{1 + 1} = \sqrt{2}
\][/tex]
The argument [tex]\( \theta \)[/tex] is:
[tex]\[
\tan(\theta) = \frac{-1}{-1} = 1 \Rightarrow \theta = -\frac{3\pi}{4}
\][/tex]
Therefore, the trigonometric form (also known as the polar form) of [tex]\( z \)[/tex] is:
[tex]\[
z = \sqrt{2} e^{-i \frac{3\pi}{4}}
\][/tex]
This completes our detailed step-by-step solution. The complex number [tex]\( z = i^{22} + i^{24n - 13} \)[/tex] in its trigonometric form is:
[tex]\[
z = \sqrt{2} e^{-i \frac{3\pi}{4}}
\][/tex]
