Questions: si los valores del triángulo son a=7.5 m ; °C=100° ; °B=47° 3.1.- determine el valor del ángulo °A 3.2.- cual es el valor del lado c del triangulo 3.3.- cual es el valor del lado b del triangulo

Solution Steps

To find angle A, we use the fact that the sum of the angles in a triangle is 180°.

\[ A + B + C = 180° \]

Given: \[ C = 100° \] \[ B = 47° \]

Substitute the given values:

\[ A + 47° + 100° = 180° \]

Solve for A:

\[ A = 180° - 147° \] \[ A = 33° \]

To find side c, we can use the Law of Sines:

\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]

Given: \[ a = 7.5 \text{ m} \] \[ A = 33° \] \[ C = 100° \]

Substitute the given values:

\[ \frac{7.5}{\sin 33°} = \frac{c}{\sin 100°} \]

Solve for c:

\[ c = \frac{7.5 \cdot \sin 100°}{\sin 33°} \]

Using a calculator to find the sine values:

\[ \sin 33° \approx 0.5446 \] \[ \sin 100° \approx 0.9848 \]

Substitute these values:

\[ c = \frac{7.5 \cdot 0.9848}{0.5446} \] \[ c \approx \frac{7.386}{0.5446} \] \[ c \approx 13.57 \text{ m} \]

To find side b, we can use the Law of Sines again:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

Given: \[ a = 7.5 \text{ m} \] \[ A = 33° \] \[ B = 47° \]

Substitute the given values:

\[ \frac{7.5}{\sin 33°} = \frac{b}{\sin 47°} \]

Solve for b:

\[ b = \frac{7.5 \cdot \sin 47°}{\sin 33°} \]

Using a calculator to find the sine values:

\[ \sin 47° \approx 0.7314 \]

Substitute these values:

\[ b = \frac{7.5 \cdot 0.7314}{0.5446} \] \[ b \approx \frac{5.4855}{0.5446} \] \[ b \approx 10.08 \text{ m} \]

Final Answer