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Law of Sines: Solving for an Angle

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Law of Sines: Solving for an Angle

Law of Sines: Solving for an Angle

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Question 1 of 5

Find

$theta$

Round your answer to the nearest minute

$theta=$ (67) $°$ (32) $'$

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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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Degree-to-Minute Conversion

$1$ degree

$=60$ minutes

Minute-to-Second Conversion

$1$ minute

$=60$ seconds

Rounding Off to the Nearest Minute

If the seconds is greater than or equal to $30”$ , round the minute up.
If the seconds is less than $30”$ , round the minute down.

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

$C$ .

$b=29 cm$

$B=theta$

$c=18 cm$

$C=35°$

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

$\frac{\color{#00880A}{29}}{\sin\color{#00880A}{\theta}}$	$=$	$\frac{\color{#9a00c7}{18}}{\sin\color{#9a00c7}{35°}}$	Substitute the values

$18timessintheta$	$=$	$29timessin35°$	Cross multiply
$18timessintheta$ $divide18$	$=$	$29timessin35°$ $divide18$	Divide both sides by $18$

$sintheta$	$=$	$(29timessin35°)/18$

$sintheta$	$=$	$16.632717/18$	Use the calculator to simplify

$sintheta$	$=$	$0.9240954$
$theta$	$=$	$sin^(-1)0.9240954$	$sin$ inverse

Simplify this further by evaluating

$sin^(-1)0.9240954$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $sin$

$3.$ Press

$0.9240954$

$4.$ Press $=$

The result will be:

$67.5323°$

Round off the answer to the nearest minute.

$theta$	$=$	$67.5323°$
	$=$	$67°31’56”$	Press DMS on your calculator
	$=$	$67°32’$	Round up since the seconds is greater than $30”$

$67°32’$

Question 2 of 5

Find

$theta$

Round your answer to the nearest degree

$theta=$ (62) $°$

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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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Degree-to-Minute Conversion

$1$ degree

$=60$ minutes

Minute-to-Second Conversion

$1$ minute

$=60$ seconds

Rounding Off to the Nearest Minute

If the seconds is greater than or equal to $30”$ , round the minute up.
If the seconds is less than $30”$ , round the minute down.

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

$p, P, r$ and

$R$ .

$p=92 m$

$P=24°$

$r=200 m$

$R=theta$

$\frac{\color{#007DDC}{p}}{\sin\color{#007DDC}{P}}$	$=$	$\frac{\color{#9a00c7}{r}}{\sin\color{#9a00c7}{R}}$

$\frac{\color{#007DDC}{92}}{\sin\color{#007DDC}{24°}}$	$=$	$\frac{\color{#9a00c7}{200}}{\sin\color{#9a00c7}{\theta}}$	Substitute the values

$92timessintheta$	$=$	$200timessin24°$	Cross multiply
$92timessintheta$ $divide92$	$=$	$200timessin24°$ $divide92$	Divide both sides by $92$

$sintheta$	$=$	$(200timessin24°)/92$

$sintheta$	$=$	$81.34733/92$	Use the calculator to simplify

$sintheta$	$=$	$0.88421$
$theta$	$=$	$sin^(-1)0.88421$	$sin$ inverse

Simplify this further by evaluating

$sin^(-1)0.88421$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $sin$

$3.$ Press

$0.88421$

$4.$ Press $=$

The result will be:

$62.15447°$

Round off the answer to the nearest degree.

$theta$	$=$	$62.15447°$
	$=$	$62°9’16”$	Press DMS on your calculator
	$=$	$62°$	Round down since the minutes is less than $30’$

$62°$

Question 3 of 5

Find

$theta$

Round your answer to the nearest degree

$theta=$ (74) $°$

Incorrect

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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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Degree-to-Minute Conversion

$1$ degree

$=60$ minutes

Minute-to-Second Conversion

$1$ minute

$=60$ seconds

Rounding Off to the Nearest Minute

If the seconds is greater than or equal to $30”$ , round the minute up.
If the seconds is less than $30”$ , round the minute down.

Since the values given are opposite each other, we can directly substitute them to the Sine Law.

$a=51.4 m$

$A=theta$

$b=52.9 m$

$B=82°$

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

$\frac{\color{#007DDC}{51.4}}{\sin\color{#007DDC}{\theta}}$	$=$	$\frac{\color{#00880A}{52.9}}{\sin\color{#00880A}{82°}}$	Substitute the values

$52.9timessintheta$	$=$	$51.4timessin82°$	Cross multiply
$52.9timessintheta$ $divide52.9$	$=$	$51.4timessin82°$ $divide52.9$	Divide both sides by $52.9$

$sintheta$	$=$	$(51.4timessin82°)/52.9$

$sintheta$	$=$	$(51.4times0.990268)/52.9$	Use the calculator to simplify

$sintheta$	$=$	$50.8997787/52.9$

$sintheta$	$=$	$0.9621886$
$theta$	$=$	$sin^(-1)0.9621886$	$sin$ inverse

Simplify this further by evaluating

$sin^(-1)0.9621886$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $sin$

$3.$ Press

$0.9621886$

$4.$ Press $=$

The result will be:

$74.1938°$

Round off the answer to the nearest degree.

$theta$	$=$	$74.1938°$
	$=$	$74°11’$	Press DMS on your calculator
	$=$	$74°$	Round down since the minutes is less than $30’$

$74°$

Question 4 of 5

Find the obtuse angle

$alpha$

Round your answer to the nearest degree

$alpha=$ (131) $°$

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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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Degree-to-Minute Conversion

$1$ degree

$=60$ minutes

Minute-to-Second Conversion

$1$ minute

$=60$ seconds

Rounding Off to the Nearest Minute

If the seconds is greater than or equal to $30”$ , round the minute up.
If the seconds is less than $30”$ , round the minute down.

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

$a=37$ m

$A=alpha$

$b=16$ m

$B=19°$

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

$\frac{\color{#007DDC}{37}}{\sin\color{#007DDC}{\alpha}}$	$=$	$\frac{\color{#00880A}{16}}{\sin\color{#00880A}{19}}$	Substitute the values

$16timessinalpha$	$=$	$37timessin19°$	Cross multiply
$16timessinalpha$ $divide16$	$=$	$37timessin19°$ $divide16$	Divide both sides by $16$

$sinalpha$	$=$	$(37timessin19°)/16$

$sinalpha$	$=$	$12.0460217/16$	Use the calculator to simplify

$sinalpha$	$=$	$0.75287636$
$alpha$	$=$	$sin^(-1)0.75287636$	$sin$ inverse

Simplify this further by evaluating

$sin^(-1)0.75287636$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $sin$

$3.$ Press

$0.75287636$

$4.$ Press $=$

The result will be:

$48.8401549°$

Round off the answer to the nearest degree.

$alpha$	$=$	$48.8401549°$
	$=$	$48°50’$	Press DMS on your calculator
	$=$	$49°$	Round up since the minutes is greater than $30’$

Notice that the result is an acute angle even though the actual angle is supposedly obtuse. This is because we used the Sine Law.

To get the obtuse angle, simply subtract the angle from

$180°$ .

$alpha$	$=$	$180-49$
$alpha$	$=$	$131°$

$131°$

Question 5 of 5

Find the obtuse angle

$alpha$

Round your answer to the nearest degree

$alpha=$ (150) $°$

Incorrect

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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

Calculator Buttons to Use

$sin$

$=$ Sine function

$cos$

$=$ Cosine function

$tan$

$=$ Tangent function

DMS or $° ‘ ‘ ‘$

$=$ Degree/Minute/Second

Shift or 2nd F or INV

$=$ Inverse function

$=$

$=$ Equal function

Degree-to-Minute Conversion

$1$ degree

$=60$ minutes

Minute-to-Second Conversion

$1$ minute

$=60$ seconds

Rounding Off to the Nearest Minute

If the seconds is greater than or equal to $30”$ , round the minute up.
If the seconds is less than $30”$ , round the minute down.

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

$a=128$ m

$A=alpha$

$b=99$ m

$B=23°$

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

$\frac{\color{#007DDC}{128}}{\sin\color{#007DDC}{\alpha}}$	$=$	$\frac{\color{#00880A}{99}}{\sin\color{#00880A}{23}}$	Substitute the values

$99timessinalpha$	$=$	$128timessin23°$	Cross multiply
$99timessinalpha$ $divide99$	$=$	$128timessin23°$ $divide99$	Divide both sides by $99$

$sinalpha$	$=$	$(128timessin23°)/99$

$sinalpha$	$=$	$50.013584/99$	Use the calculator to simplify

$sinalpha$	$=$	$0.5051877$
$alpha$	$=$	$sin^(-1)0.5051877$	$sin$ inverse

Simplify this further by evaluating

$sin^(-1)0.5051877$ using the calculator:

$1.$ Press Shift or 2nd F (depending on your calculator)

$2.$ Press $sin$

$3.$ Press

$0.5051877$

$4.$ Press $=$

The result will be:

$30.343815°$

Round off the answer to the nearest degree.

$alpha$	$=$	$30.343815°$
	$=$	$30°20’$	Press DMS on your calculator
	$=$	$30°$	Round down since the minutes is less than $30’$

Notice that the result is an acute angle even though the actual angle is supposedly obtuse. This is because we used the Sine Law.

To get the obtuse angle, simply subtract the angle from

$180°$ .

$alpha$	$=$	$180-30$
$alpha$	$=$	$150°$

$150°$