Answer:
40.8
Step-by-step explanation:
Answer:
![i^{54}=-1](https://tex.z-dn.net/?f=i%5E%7B54%7D%3D-1)
Step-by-step explanation:
First, recall the 4 basic imaginary exponents:
![i^1=i \text{, }i^2=-1\text{, }i^3=-1\text{ and } i^4=1](https://tex.z-dn.net/?f=i%5E1%3Di%20%5Ctext%7B%2C%20%7Di%5E2%3D-1%5Ctext%7B%2C%20%7Di%5E3%3D-1%5Ctext%7B%20and%20%7D%20i%5E4%3D1)
So, we want to find:
![i^{54}](https://tex.z-dn.net/?f=i%5E%7B54%7D)
This is the same as:
![=i^{52}\cdot i^2](https://tex.z-dn.net/?f=%3Di%5E%7B52%7D%5Ccdot%20i%5E2)
52 is 4 times 13. Thus:
![=(i^4)^{13}\cdot i^2](https://tex.z-dn.net/?f=%3D%28i%5E4%29%5E%7B13%7D%5Ccdot%20i%5E2)
Since we know that i to the fourth is 1:
![=(1)^{13}\cdot i^2](https://tex.z-dn.net/?f=%3D%281%29%5E%7B13%7D%5Ccdot%20i%5E2)
Simplify:
![=i^2](https://tex.z-dn.net/?f=%3Di%5E2)
And this equals:
![=-1](https://tex.z-dn.net/?f=%3D-1)
So:
![i^{54}=-1](https://tex.z-dn.net/?f=i%5E%7B54%7D%3D-1)
The answer in standard form is (x+5/2)^2+(y-2)^2=1
Finding the equivalent angle of
, the correct statements are given as follows:
- The Measure of the reference angle is 45.
.
.
<h3>What are equivalent angles?</h3>
Each angle on the second, third and fourth quadrants will have an equivalent on the first quadrant.
In this problem, the given angle is as follows:
![\theta = \frac{5\pi}{4}](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20%5Cfrac%7B5%5Cpi%7D%7B4%7D)
It is on the third quadrant, as it is between pi and 1.5 pi, hence the equivalent on the first quadrant, also known as the reference angle, is given by:
![\frac{5\pi}{4} - \pi = \frac{5\pi}{4} - \frac{4\pi}{4} = \frac{\pi}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B5%5Cpi%7D%7B4%7D%20-%20%5Cpi%20%3D%20%5Cfrac%7B5%5Cpi%7D%7B4%7D%20-%20%5Cfrac%7B4%5Cpi%7D%7B4%7D%20%3D%20%5Cfrac%7B%5Cpi%7D%7B4%7D)
The angle of 45º has equal sine and cosine, and tangent of 1, hence the correct statements are:
- The Measure of the reference angle is 45.
.
.
More can be learned about equivalent angles at brainly.com/question/24787111
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