Applying the Law of Sines - Math

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Question

In this figure, angle $a=30^\circ$ and side . If angle $b=45^\circ$ , what is the length of side ?

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Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{15}{\sin(30^\circ)}$=\frac{Y}{\sin{(45^\circ)}$}$

$$\frac{15}{\frac{1}${2}}=\frac{Y}{\frac{\sqrt{2}$}{2}}$

Cross multiply:

$15*$\frac{\sqrt{2}$}{2}=\frac{1}{2}$Y$

Multiply both sides by :

$15$\sqrt{2}$=Y$

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Question

In this figure $a=22^\circ$ and $c=85^\circ$ . If , what is ?

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Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$c}{\sin(c)}$

$$\frac{Z}{\sin(a)}$=\frac{X}{\sin{(c)}$}$

$$\frac{Z}{\sin(22^\circ)}$=\frac{30}{\sin{85^\circ}$}$

$\frac{Z}{0.375}$=\frac{30}{0.997}$

$$\frac{Z}{0.375}$=30.09$

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Question

In this figure, angle $a=30^\circ$ . If side and , what is the value of angle ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{12}{\sin(30^\circ)}$=\frac{20}{\sin{(b)}$}$

$24=\frac{20}{\sin(b)}$$

$\sin(b)=\frac{20}{24}$$

$b=56.44^\circ$

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Question

In this figure, if angle $a=18.5^\circ$ , side , and side , what is the value of angle ?

(NOTE: Figure not necessarily drawn to scale.)

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Answer

First, observe that this figure is clearly not drawn to scale. Now, we can solve using the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{30.2}{\sin(18.5^\circ)}$=\frac{17.2}{\sin{(b)}$}$

$95.18=\frac{17.2}{\sin(b)}$$

$\sin(b)=\frac{17.2}{95.18}$$

$b=10.41^\circ$

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Question

In $\Delta ABC$ , $m \angle A = $75^{\circ }$$ , $m \angle B = $62^{\circ }$$ , and . To the nearest tenth, what is ?

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Answer

Since we are given and want to find , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle A}{B C}$ = $\frac{ \sin m \angle C}{AB}$

$m \angle A = $75^{\circ }$$ and

$m \angle C = 180 - \left ( m \angle A + m \angle B \right )= 180 -(75+ 62) = 180 -137 = $43^{\circ }$$

Substitute in the above equation:

$$\frac{ $\sin75^{\circ }$$}{16} = $\frac{ \sin $43^{\circ }$$}{AB}$

Cross-multiply and solve for :

$AB\cdot $\sin75^{\circ }$ = 16 \cdot \sin $43^{\circ }$$

$AB = $\frac{16 \cdot \sin $43^{\circ }$$$}{\sin75^{\circ }$} = $\frac{16 \cdot \0.6820}{0.9659}$ \approx 11.3$

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Question

In $\Delta ABC$ , $m \angle A = $66^{\circ }$$ , , and . To the nearest tenth, what is $m \angle C$ ?

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Answer

Since we are given $m \angle A$ , , and , and want to find $m \angle C$ , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle C}{AB}$ = $\frac{ \sin m \angle A}{B C}$ .

Substitute and solve for $\sin m \angle C$ :

$$\frac{ \sin m \angle C}{16}$ = $\frac{ \sin $66^{\circ }$$}{23}$

$$\frac{ \sin m \angle C}{16}$ \cdot 16= $\frac{ \sin $66^{\circ }$$}{23}\cdot 16$

$\sin m \angle C = $\frac{ \sin $66^{\circ }$$}{23}\cdot 16 \approx $\frac{ 0.9135}{23}$\cdot 16\approx 0.6355$

Take the inverse sine of 0.6355:

$m \angle C = $\sin^{-1}$ 0.6355 \approx $39.5^{\circ }$$

There are two angles between and that have any given positive sine other than 1 - we get the other by subtracting the previous result from :

$180 - 39.5 = $140.5^{\circ }$$

This, however, is impossible, since this would result in the sum of the triangle measures being greater than . This leaves as the only possible answer.

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Question

In this figure, angle $a=30^\circ$ and side . If angle $b=45^\circ$ , what is the length of side ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{15}{\sin(30^\circ)}$=\frac{Y}{\sin{(45^\circ)}$}$

$$\frac{15}{\frac{1}${2}}=\frac{Y}{\frac{\sqrt{2}$}{2}}$

Cross multiply:

$15*$\frac{\sqrt{2}$}{2}=\frac{1}{2}$Y$

Multiply both sides by :

$15$\sqrt{2}$=Y$

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Question

In this figure $a=22^\circ$ and $c=85^\circ$ . If , what is ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$c}{\sin(c)}$

$$\frac{Z}{\sin(a)}$=\frac{X}{\sin{(c)}$}$

$$\frac{Z}{\sin(22^\circ)}$=\frac{30}{\sin{85^\circ}$}$

$\frac{Z}{0.375}$=\frac{30}{0.997}$

$$\frac{Z}{0.375}$=30.09$

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Question

In this figure, angle $a=30^\circ$ . If side and , what is the value of angle ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{12}{\sin(30^\circ)}$=\frac{20}{\sin{(b)}$}$

$24=\frac{20}{\sin(b)}$$

$\sin(b)=\frac{20}{24}$$

$b=56.44^\circ$

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Question

In this figure, if angle $a=18.5^\circ$ , side , and side , what is the value of angle ?

(NOTE: Figure not necessarily drawn to scale.)

Tap to reveal answer

Answer

First, observe that this figure is clearly not drawn to scale. Now, we can solve using the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{30.2}{\sin(18.5^\circ)}$=\frac{17.2}{\sin{(b)}$}$

$95.18=\frac{17.2}{\sin(b)}$$

$\sin(b)=\frac{17.2}{95.18}$$

$b=10.41^\circ$

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Question

In $\Delta ABC$ , $m \angle A = $75^{\circ }$$ , $m \angle B = $62^{\circ }$$ , and . To the nearest tenth, what is ?

Tap to reveal answer

Answer

Since we are given and want to find , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle A}{B C}$ = $\frac{ \sin m \angle C}{AB}$

$m \angle A = $75^{\circ }$$ and

$m \angle C = 180 - \left ( m \angle A + m \angle B \right )= 180 -(75+ 62) = 180 -137 = $43^{\circ }$$

Substitute in the above equation:

$$\frac{ $\sin75^{\circ }$$}{16} = $\frac{ \sin $43^{\circ }$$}{AB}$

Cross-multiply and solve for :

$AB\cdot $\sin75^{\circ }$ = 16 \cdot \sin $43^{\circ }$$

$AB = $\frac{16 \cdot \sin $43^{\circ }$$$}{\sin75^{\circ }$} = $\frac{16 \cdot \0.6820}{0.9659}$ \approx 11.3$

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Question

In $\Delta ABC$ , $m \angle A = $66^{\circ }$$ , , and . To the nearest tenth, what is $m \angle C$ ?

Tap to reveal answer

Answer

Since we are given $m \angle A$ , , and , and want to find $m \angle C$ , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle C}{AB}$ = $\frac{ \sin m \angle A}{B C}$ .

Substitute and solve for $\sin m \angle C$ :

$$\frac{ \sin m \angle C}{16}$ = $\frac{ \sin $66^{\circ }$$}{23}$

$$\frac{ \sin m \angle C}{16}$ \cdot 16= $\frac{ \sin $66^{\circ }$$}{23}\cdot 16$

$\sin m \angle C = $\frac{ \sin $66^{\circ }$$}{23}\cdot 16 \approx $\frac{ 0.9135}{23}$\cdot 16\approx 0.6355$

Take the inverse sine of 0.6355:

$m \angle C = $\sin^{-1}$ 0.6355 \approx $39.5^{\circ }$$

There are two angles between and that have any given positive sine other than 1 - we get the other by subtracting the previous result from :

$180 - 39.5 = $140.5^{\circ }$$

This, however, is impossible, since this would result in the sum of the triangle measures being greater than . This leaves as the only possible answer.

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Question

In this figure, angle $a=30^\circ$ and side . If angle $b=45^\circ$ , what is the length of side ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{15}{\sin(30^\circ)}$=\frac{Y}{\sin{(45^\circ)}$}$

$$\frac{15}{\frac{1}${2}}=\frac{Y}{\frac{\sqrt{2}$}{2}}$

Cross multiply:

$15*$\frac{\sqrt{2}$}{2}=\frac{1}{2}$Y$

Multiply both sides by :

$15$\sqrt{2}$=Y$

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Question

In this figure $a=22^\circ$ and $c=85^\circ$ . If , what is ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$c}{\sin(c)}$

$$\frac{Z}{\sin(a)}$=\frac{X}{\sin{(c)}$}$

$$\frac{Z}{\sin(22^\circ)}$=\frac{30}{\sin{85^\circ}$}$

$\frac{Z}{0.375}$=\frac{30}{0.997}$

$$\frac{Z}{0.375}$=30.09$

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Question

In this figure, angle $a=30^\circ$ . If side and , what is the value of angle ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{12}{\sin(30^\circ)}$=\frac{20}{\sin{(b)}$}$

$24=\frac{20}{\sin(b)}$$

$\sin(b)=\frac{20}{24}$$

$b=56.44^\circ$

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Question

In this figure, if angle $a=18.5^\circ$ , side , and side , what is the value of angle ?

(NOTE: Figure not necessarily drawn to scale.)

Tap to reveal answer

Answer

First, observe that this figure is clearly not drawn to scale. Now, we can solve using the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{30.2}{\sin(18.5^\circ)}$=\frac{17.2}{\sin{(b)}$}$

$95.18=\frac{17.2}{\sin(b)}$$

$\sin(b)=\frac{17.2}{95.18}$$

$b=10.41^\circ$

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Question

In $\Delta ABC$ , $m \angle A = $75^{\circ }$$ , $m \angle B = $62^{\circ }$$ , and . To the nearest tenth, what is ?

Tap to reveal answer

Answer

Since we are given and want to find , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle A}{B C}$ = $\frac{ \sin m \angle C}{AB}$

$m \angle A = $75^{\circ }$$ and

$m \angle C = 180 - \left ( m \angle A + m \angle B \right )= 180 -(75+ 62) = 180 -137 = $43^{\circ }$$

Substitute in the above equation:

$$\frac{ $\sin75^{\circ }$$}{16} = $\frac{ \sin $43^{\circ }$$}{AB}$

Cross-multiply and solve for :

$AB\cdot $\sin75^{\circ }$ = 16 \cdot \sin $43^{\circ }$$

$AB = $\frac{16 \cdot \sin $43^{\circ }$$$}{\sin75^{\circ }$} = $\frac{16 \cdot \0.6820}{0.9659}$ \approx 11.3$

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Question

In $\Delta ABC$ , $m \angle A = $66^{\circ }$$ , , and . To the nearest tenth, what is $m \angle C$ ?

Tap to reveal answer

Answer

Since we are given $m \angle A$ , , and , and want to find $m \angle C$ , we apply the Law of Sines, which states, in part,

$\frac{ \sin m \angle C}{AB}$ = $\frac{ \sin m \angle A}{B C}$ .

Substitute and solve for $\sin m \angle C$ :

$$\frac{ \sin m \angle C}{16}$ = $\frac{ \sin $66^{\circ }$$}{23}$

$$\frac{ \sin m \angle C}{16}$ \cdot 16= $\frac{ \sin $66^{\circ }$$}{23}\cdot 16$

$\sin m \angle C = $\frac{ \sin $66^{\circ }$$}{23}\cdot 16 \approx $\frac{ 0.9135}{23}$\cdot 16\approx 0.6355$

Take the inverse sine of 0.6355:

$m \angle C = $\sin^{-1}$ 0.6355 \approx $39.5^{\circ }$$

There are two angles between and that have any given positive sine other than 1 - we get the other by subtracting the previous result from :

$180 - 39.5 = $140.5^{\circ }$$

This, however, is impossible, since this would result in the sum of the triangle measures being greater than . This leaves as the only possible answer.

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Question

In this figure, angle $a=30^\circ$ and side . If angle $b=45^\circ$ , what is the length of side ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}$

$$\frac{Z}{\sin(a)}$=\frac{Y}{\sin{(b)}$}$

$$\frac{15}{\sin(30^\circ)}$=\frac{Y}{\sin{(45^\circ)}$}$

$$\frac{15}{\frac{1}${2}}=\frac{Y}{\frac{\sqrt{2}$}{2}}$

Cross multiply:

$15*$\frac{\sqrt{2}$}{2}=\frac{1}{2}$Y$

Multiply both sides by :

$15$\sqrt{2}$=Y$

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Question

In this figure $a=22^\circ$ and $c=85^\circ$ . If , what is ?

Tap to reveal answer

Answer

For this problem, use the law of sines:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$b}{\sin(b)}=\frac{\text{side opposite }$c}{\sin(c)}$ .

In this case, we have values that we can plug in:

$$\frac{\text{side opposite }$a}{\sin(a)}=\frac{\text{side opposite }$c}{\sin(c)}$

$$\frac{Z}{\sin(a)}$=\frac{X}{\sin{(c)}$}$

$$\frac{Z}{\sin(22^\circ)}$=\frac{30}{\sin{85^\circ}$}$

$\frac{Z}{0.375}$=\frac{30}{0.997}$

$$\frac{Z}{0.375}$=30.09$

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