Answer :
Let's solve the problem step-by-step.
### Part 1.2: Finding the terminal points
We'll start by identifying the terminal points for the given transformations using the given initial point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex] on the unit circle.
#### (a) Terminal point for [tex]\(-t + \frac{\pi}{2}\)[/tex]
For [tex]\(-t + \frac{\pi}{2}\)[/tex], the new terminal point is:
[tex]\[ \left( \cos\left(-t + \frac{\pi}{2}\right), \sin\left(-t + \frac{\pi}{2}\right) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.9230769230769231, 0.3846153846153846) \][/tex]
#### (b) Terminal point for [tex]\(t + \pi\)[/tex]
For [tex]\(t + \pi\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(t + \pi), \sin(t + \pi) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (-0.3846153846153845, 0.9230769230769231) \][/tex]
#### (c) Terminal point for [tex]\(2\pi - t\)[/tex]
For [tex]\(2\pi - t\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(2\pi - t), \sin(2\pi - t) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.38461538461538486, -0.923076923076923) \][/tex]
#### (d) Terminal point for [tex]\(t - \frac{3\pi}{2}\)[/tex]
For [tex]\(t - \frac{3\pi}{2}\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(t - \frac{3\pi}{2}), \sin(t - \frac{3\pi}{2}) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.923076923076923, 0.3846153846153848) \][/tex]
### Part 1.3: Finding the reference number [tex]\(\bar{t}\)[/tex]
Next, we'll find the reference number [tex]\(\bar{t}\)[/tex] for each given value of [tex]\(t\)[/tex].
#### (a) For [tex]\(t = \frac{7\pi}{8}\)[/tex]
The reference number is the acute angle formed between [tex]\(t\)[/tex] and the nearest x-axis. For [tex]\(\frac{7\pi}{8}\)[/tex], the reference number is:
[tex]\[ 2.748893571891069 \][/tex]
#### (b) For [tex]\(t = \frac{11\pi}{3}\)[/tex]
By reducing [tex]\(t\)[/tex] modulo [tex]\(2\pi\)[/tex], the reference number is:
[tex]\[ 1.0471975511965983 \][/tex]
#### (c) For [tex]\(t = \frac{3\pi}{2}\)[/tex]
The reference number for [tex]\(\frac{3\pi}{2}\)[/tex] is:
[tex]\[ 1.5707963267948966 \][/tex]
#### (d) For [tex]\(t = -\frac{5\pi}{6}\)[/tex]
By considering the positive equivalent of [tex]\(t\)[/tex] in the unit circle and calculating the reference number, we get:
[tex]\[ 2.6179938779914944 \][/tex]
### Final result summary
Terminal points determined:
- [tex]\(-t + \frac{\pi}{2}\)[/tex]: [tex]\((0.9230769230769231, 0.3846153846153846)\)[/tex]
- [tex]\(t + \pi\)[/tex]: [tex]\((-0.3846153846153845, 0.9230769230769231)\)[/tex]
- [tex]\(2\pi - t\)[/tex]: [tex]\((0.38461538461538486, -0.923076923076923)\)[/tex]
- [tex]\(t - \frac{3\pi}{2}\)[/tex]: [tex]\((0.923076923076923, 0.3846153846153848)\)[/tex]
Reference numbers:
- [tex]\(\frac{7\pi}{8}\)[/tex]: [tex]\(2.748893571891069\)[/tex]
- [tex]\(\frac{11\pi}{3}\)[/tex]: [tex]\(1.0471975511965983\)[/tex]
- [tex]\(\frac{3\pi}{2}\)[/tex]: [tex]\(1.5707963267948966\)[/tex]
- [tex]\(-\frac{5\pi}{6}\)[/tex]: [tex]\(2.6179938779914944\)[/tex]
### Part 1.2: Finding the terminal points
We'll start by identifying the terminal points for the given transformations using the given initial point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex] on the unit circle.
#### (a) Terminal point for [tex]\(-t + \frac{\pi}{2}\)[/tex]
For [tex]\(-t + \frac{\pi}{2}\)[/tex], the new terminal point is:
[tex]\[ \left( \cos\left(-t + \frac{\pi}{2}\right), \sin\left(-t + \frac{\pi}{2}\right) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.9230769230769231, 0.3846153846153846) \][/tex]
#### (b) Terminal point for [tex]\(t + \pi\)[/tex]
For [tex]\(t + \pi\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(t + \pi), \sin(t + \pi) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (-0.3846153846153845, 0.9230769230769231) \][/tex]
#### (c) Terminal point for [tex]\(2\pi - t\)[/tex]
For [tex]\(2\pi - t\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(2\pi - t), \sin(2\pi - t) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.38461538461538486, -0.923076923076923) \][/tex]
#### (d) Terminal point for [tex]\(t - \frac{3\pi}{2}\)[/tex]
For [tex]\(t - \frac{3\pi}{2}\)[/tex], the new terminal point is:
[tex]\[ \left( \cos(t - \frac{3\pi}{2}), \sin(t - \frac{3\pi}{2}) \right) \][/tex]
Given the original point [tex]\(\left(\frac{5}{13}, -\frac{12}{13}\right)\)[/tex], the calculations give:
[tex]\[ (0.923076923076923, 0.3846153846153848) \][/tex]
### Part 1.3: Finding the reference number [tex]\(\bar{t}\)[/tex]
Next, we'll find the reference number [tex]\(\bar{t}\)[/tex] for each given value of [tex]\(t\)[/tex].
#### (a) For [tex]\(t = \frac{7\pi}{8}\)[/tex]
The reference number is the acute angle formed between [tex]\(t\)[/tex] and the nearest x-axis. For [tex]\(\frac{7\pi}{8}\)[/tex], the reference number is:
[tex]\[ 2.748893571891069 \][/tex]
#### (b) For [tex]\(t = \frac{11\pi}{3}\)[/tex]
By reducing [tex]\(t\)[/tex] modulo [tex]\(2\pi\)[/tex], the reference number is:
[tex]\[ 1.0471975511965983 \][/tex]
#### (c) For [tex]\(t = \frac{3\pi}{2}\)[/tex]
The reference number for [tex]\(\frac{3\pi}{2}\)[/tex] is:
[tex]\[ 1.5707963267948966 \][/tex]
#### (d) For [tex]\(t = -\frac{5\pi}{6}\)[/tex]
By considering the positive equivalent of [tex]\(t\)[/tex] in the unit circle and calculating the reference number, we get:
[tex]\[ 2.6179938779914944 \][/tex]
### Final result summary
Terminal points determined:
- [tex]\(-t + \frac{\pi}{2}\)[/tex]: [tex]\((0.9230769230769231, 0.3846153846153846)\)[/tex]
- [tex]\(t + \pi\)[/tex]: [tex]\((-0.3846153846153845, 0.9230769230769231)\)[/tex]
- [tex]\(2\pi - t\)[/tex]: [tex]\((0.38461538461538486, -0.923076923076923)\)[/tex]
- [tex]\(t - \frac{3\pi}{2}\)[/tex]: [tex]\((0.923076923076923, 0.3846153846153848)\)[/tex]
Reference numbers:
- [tex]\(\frac{7\pi}{8}\)[/tex]: [tex]\(2.748893571891069\)[/tex]
- [tex]\(\frac{11\pi}{3}\)[/tex]: [tex]\(1.0471975511965983\)[/tex]
- [tex]\(\frac{3\pi}{2}\)[/tex]: [tex]\(1.5707963267948966\)[/tex]
- [tex]\(-\frac{5\pi}{6}\)[/tex]: [tex]\(2.6179938779914944\)[/tex]
