Question

solve each triangle. round your answers to the nearest tenth. 1) (mangle b = 25^{circ},mangle c = 127^{circ},b = 9) (mangle a = 28^{circ},c = 17,a = 10) 3) (c = 18mathrm{cm},mangle a = 137^{circ},b = 12mathrm{cm}) (mangle b = 17^{circ},mangle c = 26^{circ},a = 28mathrm{cm})

Explanation:

Step1: Find angle A in the first triangle

The sum of angles in a triangle is 180°. So, $m\angle A=180^{\circ}-m\angle B - m\angle C$. Given $m\angle B = 25^{\circ}$ and $m\angle C=127^{\circ}$, then $m\angle A=180^{\circ}-25^{\circ}-127^{\circ}=28^{\circ}$.

Step2: Use the Law of Sines to find side a

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}$. We know $b = 9$, $m\angle A = 28^{\circ}$, and $m\angle B=25^{\circ}$. So, $a=\frac{b\sin A}{\sin B}=\frac{9\times\sin28^{\circ}}{\sin25^{\circ}}$. Using a calculator, $\sin28^{\circ}\approx0.4695$ and $\sin25^{\circ}\approx0.4226$. Then $a=\frac{9\times0.4695}{0.4226}\approx10.0$.

Step3: Use the Law of Sines to find side c

Again, by the Law of Sines $\frac{c}{\sin C}=\frac{b}{\sin B}$. So, $c=\frac{b\sin C}{\sin B}=\frac{9\times\sin127^{\circ}}{\sin25^{\circ}}$. Since $\sin127^{\circ}\approx0.7986$, then $c=\frac{9\times0.7986}{0.4226}\approx17.0$.

For the third - triangle:

Step1: Find angle C

$m\angle C=180^{\circ}-m\angle A - m\angle B$. Given $m\angle A = 137^{\circ}$ and $m\angle B = 17^{\circ}$, then $m\angle C=180^{\circ}-137^{\circ}-17^{\circ}=26^{\circ}$.

Step2: Use the Law of Sines to find side a

By the Law of Sines $\frac{a}{\sin A}=\frac{b}{\sin B}$. We know $b = 12$, $m\angle A=137^{\circ}$, and $m\angle B = 17^{\circ}$. So, $a=\frac{b\sin A}{\sin B}=\frac{12\times\sin137^{\circ}}{\sin17^{\circ}}$. Since $\sin137^{\circ}\approx0.6820$ and $\sin17^{\circ}\approx0.2924$, then $a=\frac{12\times0.6820}{0.2924}\approx28.0$.

Step3: Use the Law of Sines to find side c

By the Law of Sines $\frac{c}{\sin C}=\frac{b}{\sin B}$. So, $c=\frac{b\sin C}{\sin B}=\frac{12\times\sin26^{\circ}}{\sin17^{\circ}}$. Since $\sin26^{\circ}\approx0.4384$, then $c=\frac{12\times0.4384}{0.2924}\approx18.0$.

Answer:

For the first triangle: $m\angle A = 28^{\circ},a\approx10.0,c\approx17.0$
For the third triangle: $m\angle C = 26^{\circ},a\approx28.0,c\approx18.0$