Law of Sines: Solving for a Side

Question 1 of 6

Find

$x$

Round your answer to

$1$ decimal place

$x=$ (23.7) $m$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

Substitute the three known values to the Sine Law to find the fourth missing value.

From labelling the triangle, we know that the known values are those with labels

$a, A, b$ and

$B$ .

$a=x$

$A=51°$

$b=29 m$

$B=72°$

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}$	$=$	$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}$

$\frac{\color{#007DDC}{x}}{\sin\color{#007DDC}{51°}}$	$=$	$\frac{\color{#00880A}{29}}{\sin\color{#00880A}{72°}}$	Substitute the values

$xtimessin72°$	$=$	$29timessin51°$	Cross multiply
$xtimessin72°$ $dividesin72°$	$=$	$29timessin51°$ $dividesin72°$	Divide both sides by $sin72°$

$x$	$=$	$(29timessin51°)/(sin72°)$

$x$	$=$	$22.53723/0.9510565$	Use the calculator to simplify

$x$	$=$	$23.697$
$x$	$=$	$23.7 m$	Rounded off to $1$ decimal place

$23.7 m$

Question 2 of 6

Find

$x$

Round your answer to

$1$ decimal place

$x=$ (141.4) $cm$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

Substitute the three known values to the Sine Law to find the fourth missing value.

From labelling the triangle, we know that the known values are those with labels

$a, A, c$ and

$C$ .

$a=x$

$A=63°$

$c=130 cm$

$C=55°$

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}$	$=$	$\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

$\frac{\color{#007DDC}{x}}{\sin\color{#007DDC}{63°}}$	$=$	$\frac{\color{#9a00c7}{130}}{\sin\color{#9a00c7}{55°}}$	Substitute the values

$xtimessin55°$	$=$	$130timessin63°$	Cross multiply
$xtimessin55°$ $dividesin55°$	$=$	$130timessin63°$ $dividesin55°$	Divide both sides by $sin55°$

$x$	$=$	$(130timessin63°)/(sin55°)$

$x$	$=$	$115.830848/0.81915204$	Use the calculator to simplify

$x$	$=$	$141.4033$
$x$	$=$	$141.4 cm$	Rounded off to $1$ decimal place

$141.4 cm$

Question 3 of 6

Find

$c$

Round your answer to

$1$ decimal place

$c=$ (80.9) $m$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

Substitute the three known values to the Sine Law to find the fourth missing value.

From labelling the triangle, we know that the known values are those with labels

$b, B, c$ and

$C$ .

$b=62.3 m$

$B=50°$

$c=c$

$C=84°$

$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}$	$=$	$\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

$\frac{\color{#00880A}{62.3}}{\sin\color{#00880A}{50°}}$	$=$	$\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{84°}}$	Substitute the values

$ctimessin50°$	$=$	$62.3timessin84°$	Cross multiply
$ctimessin50°$ $dividesin50°$	$=$	$62.3timessin84°$ $dividesin50°$	Divide both sides by $sin50°$

$c$	$=$	$(62.3timessin84°)/(sin50°)$

$c$	$=$	$61.958714/0.76604444$	Use the calculator to simplify

$c$	$=$	$80.88$
$c$	$=$	$80.9 m$	Rounded off to $1$ decimal place

$80.9 m$

Question 4 of 6

Find

$b$

Round your answer to

$2$ decimal places

$b=$ (85.07) $cm$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

Substitute the three known values to the Sine Law to find the fourth missing value.

From labelling the triangle, we know that the known values are those with labels

$b, B, c$ and

$C$ .

$b=b$

$B=116°$

$c=40 cm$

$C=25°$

$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}$	$=$	$\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{116°}}$	$=$	$\frac{\color{#9a00c7}{40}}{\sin\color{#9a00c7}{25°}}$	Substitute the values

$btimessin25°$	$=$	$40timessin116°$	Cross multiply
$btimessin25°$ $dividesin25°$	$=$	$40timessin116°$ $dividesin25°$	Divide both sides by $sin25°$

$b$	$=$	$(40timessin116°)/(sin25°)$

$b$	$=$	$35.95176/0.422618$	Use the calculator to simplify

$b$	$=$	$85.069$
$b$	$=$	$85.07 cm$	Rounded off to $2$ decimal places

$85.07 cm$

Question 5 of 6

Find

$LN$

Round your answer to

$1$ decimal place

$LN=$ (251.5) $cm$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

Since the length of side

$n$ is given, it’s best to use it instead of

$m$ .

Solve for angle

$N$ knowing that the sum of all interior angles in a triangle is

$180°$ .

$N$	$=$	$180°-(133°+32°)$
	$=$	$180°-165°$
	$=$	$15°$

Substitute

$m, M, n$ and

$N$ to the Sine Law to find

$m$ or

$LN$ .

$m=m$ or

$LN$

$M=133°$

$n=89 cm$

$N=15°$

$\frac{\color{#00880A}{m}}{\sin\color{#00880A}{M}}$	$=$	$\frac{\color{#9a00c7}{n}}{\sin\color{#9a00c7}{N}}$

$\frac{\color{#00880A}{m}}{\sin\color{#00880A}{133°}}$	$=$	$\frac{\color{#9a00c7}{89}}{\sin\color{#9a00c7}{15°}}$	Substitute the values

$mtimessin15°$	$=$	$89timessin133°$	Cross multiply
$mtimessin15°$ $dividesin15°$	$=$	$89timessin133°$ $dividesin15°$	Divide both sides by $sin15°$

$m$	$=$	$(89timessin133°)/(sin15°)$

$m$	$=$	$65.09048/0.258819$	Use the calculator to simplify

$m$	$=$	$251.49$
$m$ or $LN$	$=$	$251.5 cm$	Rounded off to $1$ decimal place

$251.5 cm$

Question 6 of 6

Find

$AC$

Round your answer to

$1$ decimal place

$AC=$ (35.8) $km$

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Sine Law

$\frac{\color{#007DDC}{a}}{\sin\color{#007DDC}{A}}=\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}=\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

where:
$a$ is the side opposite angle $A$
$b$ is the side opposite angle $B$
$c$ is the side opposite angle $C$

When to use the Sine Law (for non-right angled triangles)

a) Given 2 sides and 1 angle to find the other angle

or

b) Given 2 angles 1 side to find the other side

First, label the triangle according to the Sine Law.

To solve for side

$AC$ or

$b$ , angle

$B$ must be calculated.

Solve for angle

$B$ knowing that the sum of all interior angles in a triangle is

$180°$ .

$B$	$=$	$180°-(76°+24°)$
	$=$	$180°-100°$
	$=$	$80°$

Substitute

$b, B, c$ and

$C$ to the Sine Law to find

$b$ or

$AC$ .

$b=AC$

$B=80°$

$c=14.8 km$

$C=24°$

$\frac{\color{#00880A}{b}}{\sin\color{#00880A}{B}}$	$=$	$\frac{\color{#9a00c7}{c}}{\sin\color{#9a00c7}{C}}$

$\frac{\color{#00880A}{AC}}{\sin\color{#00880A}{80°}}$	$=$	$\frac{\color{#9a00c7}{14.8}}{\sin\color{#9a00c7}{24°}}$	Substitute the values

$ACtimessin24°$	$=$	$14.8timessin80°$	Cross multiply
$ACtimessin24°$ $dividesin24°$	$=$	$14.8timessin80°$ $dividesin24°$	Divide both sides by $sin24°$

$AC$	$=$	$(14.8timessin80°)/(sin24°)$

$AC$	$=$	$(14.8times0.9848)/(0.4067366)$	Use the calculator to simplify

$AC$	$=$	$35.83438$
$AC$	$=$	$35.8 km$	Rounded off to $1$ decimal place

$35.8 km$

Law of Sines: Solving for a Side

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Quiz summary

Information

1. Question

Hint

Fantastic!

Sine Law

When to use the Sine Law (for non-right angled triangles)

2. Question

Hint

Well Done!

Sine Law

When to use the Sine Law (for non-right angled triangles)

3. Question

Hint

Correct!

Sine Law

When to use the Sine Law (for non-right angled triangles)

4. Question

Hint

Great Work!

Sine Law

When to use the Sine Law (for non-right angled triangles)

5. Question

Hint

Fantastic!

Sine Law

When to use the Sine Law (for non-right angled triangles)

6. Question

Hint

Excellent!

Sine Law

When to use the Sine Law (for non-right angled triangles)

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