Answer :
Para resolver el problema de encontrar las funciones trigonométricas para los ángulos dados, aproximadas a cuatro decimales, seguimos los siguientes pasos:
### Paso 1: Convertir los ángulos en forma de grados, minutos y segundos a grados decimales cuando sea necesario.
1. Ángulo [tex]\(b\)[/tex]: 192° 22' 19"
[tex]\[192° + \frac{22'}{60} + \frac{19"}{3600} = 192.372 \text{ grados}\][/tex]
2. Ángulo [tex]\(c\)[/tex]: -45° 10'
[tex]\[-45° - \frac{10'}{60} = -45.1667 \text{ grados}\][/tex]
3. Ángulo [tex]\(d\)[/tex]: 223° 3' 35"
[tex]\[223° + \frac{3'}{60} + \frac{35"}{3600} = 223.0597 \text{ grados}\][/tex]
4. Ángulo [tex]\(g\)[/tex]: -300° 25'
[tex]\[-300° - \frac{25'}{60} = -300.4167 \text{ grados}\][/tex]
5. Ángulo [tex]\(i\)[/tex]: 121° 5' 47"
[tex]\[121° + \frac{5'}{60} + \frac{47"}{3600} = 121.0964 \text{ grados}\][/tex]
6. Ángulo [tex]\(j\)[/tex]: 215° 18' 5"
[tex]\[215° + \frac{18'}{60} + \frac{5"}{3600} = 215.3014 \text{ grados}\][/tex]
### Paso 2: Calcular las funciones trigonométricas (seno, coseno y tangente) para cada ángulo.
1. Ángulo [tex]\(a\)[/tex]: 115°
- [tex]\(\sin(115°) = 0.9063\)[/tex]
- [tex]\(\cos(115°) = -0.4226\)[/tex]
- [tex]\(\tan(115°) = -2.1445\)[/tex]
2. Ángulo [tex]\(b\)[/tex]: 192.372°
- [tex]\(\sin(192.372°) = -0.2143\)[/tex]
- [tex]\(\cos(192.372°) = -0.9768\)[/tex]
- [tex]\(\tan(192.372°) = 0.2194\)[/tex]
3. Ángulo [tex]\(f\)[/tex]: 300°
- [tex]\(\sin(300°) = -0.8660\)[/tex]
- [tex]\(\cos(300°) = 0.5000\)[/tex]
- [tex]\(\tan(300°) = -1.7321\)[/tex]
4. Ángulo [tex]\(g\)[/tex]: -300.4167°
- [tex]\(\sin(-300.4167°) = 0.8624\)[/tex]
- [tex]\(\cos(-300.4167°) = 0.5063\)[/tex]
- [tex]\(\tan(-300.4167°) = 1.7033\)[/tex]
5. Ángulo [tex]\(k\)[/tex]: -215°
- [tex]\(\sin(-215°) = 0.5736\)[/tex]
- [tex]\(\cos(-215°) = -0.8192\)[/tex]
- [tex]\(\tan(-215°) = -0.7002\)[/tex]
6. Ángulo [tex]\(l\)[/tex]: -285°
- [tex]\(\sin(-285°) = 0.9659\)[/tex]
- [tex]\(\cos(-285°) = 0.2588\)[/tex]
- [tex]\(\tan(-285°) = 3.7321\)[/tex]
7. Ángulo [tex]\(c\)[/tex]: -45.1667°
- [tex]\(\sin(-45.1667°) = -0.7092\)[/tex]
- [tex]\(\cos(-45.1667°) = 0.7050\)[/tex]
- [tex]\(\tan(-45.1667°) = -1.0058\)[/tex]
8. Ángulo [tex]\(h\)[/tex]: 45°
- [tex]\(\sin(45°) = 0.7071\)[/tex]
- [tex]\(\cos(45°) = 0.7071\)[/tex]
- [tex]\(\tan(45°) = 1.0000\)[/tex]
9. Ángulo [tex]\(m\)[/tex]: -120°
- [tex]\(\sin(-120°) = -0.8660\)[/tex]
- [tex]\(\cos(-120°) = -0.5000\)[/tex]
- [tex]\(\tan(-120°) = 1.7321\)[/tex]
10. Ángulo [tex]\(d\)[/tex]: 223.0597°
- [tex]\(\sin(223.0597°) = -0.6828\)[/tex]
- [tex]\(\cos(223.0597°) = -0.7306\)[/tex]
- [tex]\(\tan(223.0597°) = 0.9345\)[/tex]
11. Ángulo [tex]\(i\)[/tex]: 121.0964°
- [tex]\(\sin(121.0964°) = 0.8563\)[/tex]
- [tex]\(\cos(121.0964°) = -0.5165\)[/tex]
- [tex]\(\tan(121.0964°) = -1.6580\)[/tex]
12. Ángulo [tex]\(n\)[/tex]: 60°
- [tex]\(\sin(60°) = 0.8660\)[/tex]
- [tex]\(\cos(60°) = 0.5000\)[/tex]
- [tex]\(\tan(60°) = 1.7321\)[/tex]
13. Ángulo [tex]\(e\)[/tex]: 180°
- [tex]\(\sin(180°) = 0.0000\)[/tex]
- [tex]\(\cos(180°) = -1.0000\)[/tex]
- [tex]\(\tan(180°) = -0.0000\)[/tex]
14. Ángulo [tex]\(j\)[/tex]: 215.3014°
- [tex]\(\sin(215.3014°) = -0.5779\)[/tex]
- [tex]\(\cos(215.3014°) = -0.8161\)[/tex]
- [tex]\(\tan(215.3014°) = 0.7081\)[/tex]
### Resumen de resultados:
- 115°:
- [tex]\(\sin = 0.9063\)[/tex]
- [tex]\(\cos = -0.4226\)[/tex]
- [tex]\(\tan = -2.1445\)[/tex]
- 192° 22' 19":
- [tex]\(\sin = -0.2143\)[/tex]
- [tex]\(\cos = -0.9768\)[/tex]
- [tex]\(\tan = 0.2194\)[/tex]
- 300°:
- [tex]\(\sin = -0.8660\)[/tex]
- [tex]\(\cos = 0.5000\)[/tex]
- [tex]\(\tan = -1.7321\)[/tex]
- -300° 25':
- [tex]\(\sin = 0.8624\)[/tex]
- [tex]\(\cos = 0.5063\)[/tex]
- [tex]\(\tan = 1.7033\)[/tex]
- -215°:
- [tex]\(\sin = 0.5736\)[/tex]
- [tex]\(\cos = -0.8192\)[/tex]
- [tex]\(\tan = -0.7002\)[/tex]
- -285°:
- [tex]\(\sin = 0.9659\)[/tex]
- [tex]\(\cos = 0.2588\)[/tex]
- [tex]\(\tan = 3.7321\)[/tex]
- -45° 10':
- [tex]\(\sin = -0.7092\)[/tex]
- [tex]\(\cos = 0.7050\)[/tex]
- [tex]\(\tan = -1.0058\)[/tex]
- 45°:
- [tex]\(\sin = 0.7071\)[/tex]
- [tex]\(\cos = 0.7071\)[/tex]
- [tex]\(\tan = 1.0000\)[/tex]
- -120°:
- [tex]\(\sin = -0.8660\)[/tex]
- [tex]\(\cos = -0.5000\)[/tex]
- [tex]\(\tan = 1.7321\)[/tex]
- 223° 3' 35":
- [tex]\(\sin = -0.6828\)[/tex]
- [tex]\(\cos = -0.7306\)[/tex]
- [tex]\(\tan = 0.9345\)[/tex]
- 121° 5' 47":
- [tex]\(\sin = 0.8563\)[/tex]
- [tex]\(\cos = -0.5165\)[/tex]
- [tex]\(\tan = -1.6580\)[/tex]
- 60°:
- [tex]\(\sin = 0.8660\)[/tex]
- [tex]\(\cos = 0.5000\)[/tex]
- [tex]\(\tan = 1.7321\)[/tex]
- 180°:
- [tex]\(\sin = 0.0000\)[/tex]
- [tex]\(\cos = -1.0000\)[/tex]
- [tex]\(\tan = -0.0000\)[/tex]
- 215° 18' 5":
- [tex]\(\sin = -0.5779\)[/tex]
- [tex]\(\cos = -0.8161\)[/tex]
- [tex]\(\tan = 0.7081\)[/tex]
### Paso 1: Convertir los ángulos en forma de grados, minutos y segundos a grados decimales cuando sea necesario.
1. Ángulo [tex]\(b\)[/tex]: 192° 22' 19"
[tex]\[192° + \frac{22'}{60} + \frac{19"}{3600} = 192.372 \text{ grados}\][/tex]
2. Ángulo [tex]\(c\)[/tex]: -45° 10'
[tex]\[-45° - \frac{10'}{60} = -45.1667 \text{ grados}\][/tex]
3. Ángulo [tex]\(d\)[/tex]: 223° 3' 35"
[tex]\[223° + \frac{3'}{60} + \frac{35"}{3600} = 223.0597 \text{ grados}\][/tex]
4. Ángulo [tex]\(g\)[/tex]: -300° 25'
[tex]\[-300° - \frac{25'}{60} = -300.4167 \text{ grados}\][/tex]
5. Ángulo [tex]\(i\)[/tex]: 121° 5' 47"
[tex]\[121° + \frac{5'}{60} + \frac{47"}{3600} = 121.0964 \text{ grados}\][/tex]
6. Ángulo [tex]\(j\)[/tex]: 215° 18' 5"
[tex]\[215° + \frac{18'}{60} + \frac{5"}{3600} = 215.3014 \text{ grados}\][/tex]
### Paso 2: Calcular las funciones trigonométricas (seno, coseno y tangente) para cada ángulo.
1. Ángulo [tex]\(a\)[/tex]: 115°
- [tex]\(\sin(115°) = 0.9063\)[/tex]
- [tex]\(\cos(115°) = -0.4226\)[/tex]
- [tex]\(\tan(115°) = -2.1445\)[/tex]
2. Ángulo [tex]\(b\)[/tex]: 192.372°
- [tex]\(\sin(192.372°) = -0.2143\)[/tex]
- [tex]\(\cos(192.372°) = -0.9768\)[/tex]
- [tex]\(\tan(192.372°) = 0.2194\)[/tex]
3. Ángulo [tex]\(f\)[/tex]: 300°
- [tex]\(\sin(300°) = -0.8660\)[/tex]
- [tex]\(\cos(300°) = 0.5000\)[/tex]
- [tex]\(\tan(300°) = -1.7321\)[/tex]
4. Ángulo [tex]\(g\)[/tex]: -300.4167°
- [tex]\(\sin(-300.4167°) = 0.8624\)[/tex]
- [tex]\(\cos(-300.4167°) = 0.5063\)[/tex]
- [tex]\(\tan(-300.4167°) = 1.7033\)[/tex]
5. Ángulo [tex]\(k\)[/tex]: -215°
- [tex]\(\sin(-215°) = 0.5736\)[/tex]
- [tex]\(\cos(-215°) = -0.8192\)[/tex]
- [tex]\(\tan(-215°) = -0.7002\)[/tex]
6. Ángulo [tex]\(l\)[/tex]: -285°
- [tex]\(\sin(-285°) = 0.9659\)[/tex]
- [tex]\(\cos(-285°) = 0.2588\)[/tex]
- [tex]\(\tan(-285°) = 3.7321\)[/tex]
7. Ángulo [tex]\(c\)[/tex]: -45.1667°
- [tex]\(\sin(-45.1667°) = -0.7092\)[/tex]
- [tex]\(\cos(-45.1667°) = 0.7050\)[/tex]
- [tex]\(\tan(-45.1667°) = -1.0058\)[/tex]
8. Ángulo [tex]\(h\)[/tex]: 45°
- [tex]\(\sin(45°) = 0.7071\)[/tex]
- [tex]\(\cos(45°) = 0.7071\)[/tex]
- [tex]\(\tan(45°) = 1.0000\)[/tex]
9. Ángulo [tex]\(m\)[/tex]: -120°
- [tex]\(\sin(-120°) = -0.8660\)[/tex]
- [tex]\(\cos(-120°) = -0.5000\)[/tex]
- [tex]\(\tan(-120°) = 1.7321\)[/tex]
10. Ángulo [tex]\(d\)[/tex]: 223.0597°
- [tex]\(\sin(223.0597°) = -0.6828\)[/tex]
- [tex]\(\cos(223.0597°) = -0.7306\)[/tex]
- [tex]\(\tan(223.0597°) = 0.9345\)[/tex]
11. Ángulo [tex]\(i\)[/tex]: 121.0964°
- [tex]\(\sin(121.0964°) = 0.8563\)[/tex]
- [tex]\(\cos(121.0964°) = -0.5165\)[/tex]
- [tex]\(\tan(121.0964°) = -1.6580\)[/tex]
12. Ángulo [tex]\(n\)[/tex]: 60°
- [tex]\(\sin(60°) = 0.8660\)[/tex]
- [tex]\(\cos(60°) = 0.5000\)[/tex]
- [tex]\(\tan(60°) = 1.7321\)[/tex]
13. Ángulo [tex]\(e\)[/tex]: 180°
- [tex]\(\sin(180°) = 0.0000\)[/tex]
- [tex]\(\cos(180°) = -1.0000\)[/tex]
- [tex]\(\tan(180°) = -0.0000\)[/tex]
14. Ángulo [tex]\(j\)[/tex]: 215.3014°
- [tex]\(\sin(215.3014°) = -0.5779\)[/tex]
- [tex]\(\cos(215.3014°) = -0.8161\)[/tex]
- [tex]\(\tan(215.3014°) = 0.7081\)[/tex]
### Resumen de resultados:
- 115°:
- [tex]\(\sin = 0.9063\)[/tex]
- [tex]\(\cos = -0.4226\)[/tex]
- [tex]\(\tan = -2.1445\)[/tex]
- 192° 22' 19":
- [tex]\(\sin = -0.2143\)[/tex]
- [tex]\(\cos = -0.9768\)[/tex]
- [tex]\(\tan = 0.2194\)[/tex]
- 300°:
- [tex]\(\sin = -0.8660\)[/tex]
- [tex]\(\cos = 0.5000\)[/tex]
- [tex]\(\tan = -1.7321\)[/tex]
- -300° 25':
- [tex]\(\sin = 0.8624\)[/tex]
- [tex]\(\cos = 0.5063\)[/tex]
- [tex]\(\tan = 1.7033\)[/tex]
- -215°:
- [tex]\(\sin = 0.5736\)[/tex]
- [tex]\(\cos = -0.8192\)[/tex]
- [tex]\(\tan = -0.7002\)[/tex]
- -285°:
- [tex]\(\sin = 0.9659\)[/tex]
- [tex]\(\cos = 0.2588\)[/tex]
- [tex]\(\tan = 3.7321\)[/tex]
- -45° 10':
- [tex]\(\sin = -0.7092\)[/tex]
- [tex]\(\cos = 0.7050\)[/tex]
- [tex]\(\tan = -1.0058\)[/tex]
- 45°:
- [tex]\(\sin = 0.7071\)[/tex]
- [tex]\(\cos = 0.7071\)[/tex]
- [tex]\(\tan = 1.0000\)[/tex]
- -120°:
- [tex]\(\sin = -0.8660\)[/tex]
- [tex]\(\cos = -0.5000\)[/tex]
- [tex]\(\tan = 1.7321\)[/tex]
- 223° 3' 35":
- [tex]\(\sin = -0.6828\)[/tex]
- [tex]\(\cos = -0.7306\)[/tex]
- [tex]\(\tan = 0.9345\)[/tex]
- 121° 5' 47":
- [tex]\(\sin = 0.8563\)[/tex]
- [tex]\(\cos = -0.5165\)[/tex]
- [tex]\(\tan = -1.6580\)[/tex]
- 60°:
- [tex]\(\sin = 0.8660\)[/tex]
- [tex]\(\cos = 0.5000\)[/tex]
- [tex]\(\tan = 1.7321\)[/tex]
- 180°:
- [tex]\(\sin = 0.0000\)[/tex]
- [tex]\(\cos = -1.0000\)[/tex]
- [tex]\(\tan = -0.0000\)[/tex]
- 215° 18' 5":
- [tex]\(\sin = -0.5779\)[/tex]
- [tex]\(\cos = -0.8161\)[/tex]
- [tex]\(\tan = 0.7081\)[/tex]
