Question

1. solve the triangle. in △def, <e is 52°, e = 25 cm and f = 28 cm

Explanation:

Step1: Use the Law of Sines to find angle F

By the Law of Sines, $\frac{\sin F}{f}=\frac{\sin E}{e}$. Substitute $E = 52^{\circ}$, $e = 25$ cm and $f = 28$ cm. So $\sin F=\frac{f\sin E}{e}=\frac{28\sin52^{\circ}}{25}$.
$\sin F=\frac{28\times0.788}{25}=\frac{22.064}{25}=0.88256$. Then $F=\sin^{- 1}(0.88256)\approx61.9^{\circ}$ or $F = 180^{\circ}-61.9^{\circ}=118.1^{\circ}$.

Step2: Find angle D for the first - case of F

If $F\approx61.9^{\circ}$, since the sum of angles in a triangle is $180^{\circ}$, then $D=180^{\circ}-E - F=180^{\circ}-52^{\circ}-61.9^{\circ}=66.1^{\circ}$.

Step3: Use the Law of Sines to find side d for the first - case of F

Again, by the Law of Sines $\frac{d}{\sin D}=\frac{e}{\sin E}$. So $d=\frac{e\sin D}{\sin E}=\frac{25\sin66.1^{\circ}}{\sin52^{\circ}}=\frac{25\times0.914}{0.788}\approx29.0$ cm.

Step4: Find angle D for the second - case of F

If $F = 118.1^{\circ}$, then $D=180^{\circ}-E - F=180^{\circ}-52^{\circ}-118.1^{\circ}=9.9^{\circ}$.

Step5: Use the Law of Sines to find side d for the second - case of F

By the Law of Sines $\frac{d}{\sin D}=\frac{e}{\sin E}$. So $d=\frac{e\sin D}{\sin E}=\frac{25\sin9.9^{\circ}}{\sin52^{\circ}}=\frac{25\times0.172}{0.788}\approx5.4$ cm.

Two possible solutions:
Case 1: $F\approx61.9^{\circ}$, $D = 66.1^{\circ}$, $d\approx29.0$ cm
Case 2: $F = 118.1^{\circ}$, $D=9.9^{\circ}$, $d\approx5.4$ cm

Answer:

Case 1: $F\approx61.9^{\circ}$, $D = 66.1^{\circ}$, $d\approx29.0$ cm; Case 2: $F = 118.1^{\circ}$, $D=9.9^{\circ}$, $d\approx5.4$ cm