Certainly. Let's determine the exact values of [tex]\(\sin \theta\)[/tex], [tex]\(\cos \theta\)[/tex], and [tex]\(\tan \theta\)[/tex] for each given point.
### a) [tex]\(P (-5, 12)\)[/tex]
1. Radius ([tex]\(r\)[/tex]) Calculation:
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\][/tex]
2. Sine ([tex]\(\sin \theta\)[/tex]) Calculation:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{12}{13} \approx 0.923
\][/tex]
3. Cosine ([tex]\(\cos \theta\)[/tex]) Calculation:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{-5}{13} \approx -0.385
\][/tex]
4. Tangent ([tex]\(\tan \theta\)[/tex]) Calculation:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{12}{-5} = -2.4
\][/tex]
The values are:
[tex]\[
\sin \theta \approx 0.923, \quad \cos \theta \approx -0.385, \quad \tan \theta = -2.4
\][/tex]
### b) [tex]\(P (5, -3)\)[/tex]
1. Radius ([tex]\(r\)[/tex]) Calculation:
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
\][/tex]
2. Sine ([tex]\(\sin \theta\)[/tex]) Calculation:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{-3}{\sqrt{34}} \approx -0.514
\][/tex]
3. Cosine ([tex]\(\cos \theta\)[/tex]) Calculation:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{5}{\sqrt{34}} \approx 0.857
\][/tex]
4. Tangent ([tex]\(\tan \theta\)[/tex]) Calculation:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{-3}{5} = -0.6
\][/tex]
The values are:
[tex]\[
\sin \theta \approx -0.514, \quad \cos \theta \approx 0.857, \quad \tan \theta = -0.6
\][/tex]
### c) [tex]\(P (6, 3)\)[/tex]
1. Radius ([tex]\(r\)[/tex]) Calculation:
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\][/tex]
2. Sine ([tex]\(\sin \theta\)[/tex]) Calculation:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{3}{3\sqrt{5}} = \frac{1}{\sqrt{5}} \approx 0.447
\][/tex]
3. Cosine ([tex]\(\cos \theta\)[/tex]) Calculation:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{6}{3\sqrt{5}} = \frac{2}{\sqrt{5}} \approx 0.894
\][/tex]
4. Tangent ([tex]\(\tan \theta\)[/tex]) Calculation:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{3}{6} = 0.5
\][/tex]
The values are:
[tex]\[
\sin \theta \approx 0.447, \quad \cos \theta \approx 0.894, \quad \tan \theta = 0.5
\][/tex]
### d) [tex]\(P (-24, -10)\)[/tex]
1. Radius ([tex]\(r\)[/tex]) Calculation:
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{(-24)^2 + (-10)^2} = \sqrt{576 + 100} = \sqrt{676} = 26
\][/tex]
2. Sine ([tex]\(\sin \theta\)[/tex]) Calculation:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{-10}{26} \approx -0.385
\][/tex]
3. Cosine ([tex]\(\cos \theta\)[/tex]) Calculation:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{-24}{26} \approx -0.923
\][/tex]
4. Tangent ([tex]\(\tan \theta\)[/tex]) Calculation:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{-10}{-24} \approx 0.417
\][/tex]
The values are:
[tex]\[
\sin \theta \approx -0.385, \quad \cos \theta \approx -0.923, \quad \tan \theta \approx 0.417
\][/tex]
