Cosine Rule: Solving for an Angle

Question 1 of 5

Find

θ

$theta$

Round your answer to the nearest degree

$θ =$ $theta=$ (121) $°$ $°$

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Cosine Rule

a^{2} = b^{2} + c^{2} - 2 b c cos A

$\color{#007DDC}{a}^2=\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-2\color{#00880A}{b}\color{#9a00c7}{c}\cos\color{#007DDC}{A}$

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

When to use the Cosine Rule (for non-right angled triangles)

a) Given 3 sides to find an angle

or

b) Given 2 sides and 1 angle to find the other side

Calculator Buttons to Use

$sin$ $sin$

=

$=$ Sine function

$cos$ $cos$

=

$=$ Cosine function

$tan$ $tan$

=

$=$ Tangent function

DMS or $° ‘ ‘ ‘$ $° ‘ ‘ ‘$

=

$=$ Degree/Minute/Second

Shift or 2nd F or INV

=

$=$ Inverse function

$=$ $=$

=

$=$ Equal function

Since

3

$3$ sides are given, use the Cosine Rule.

First, label the triangle according to the Cosine Rule.

Substitute the three known values to the Cosine Rule to find

θ

$theta$ or $A$ $A$ .

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

a, b

$a, b$ and

c

$c$ .

A = θ

$A=theta$

a = 15 c m

$a=15 cm$

b = 11 c m

$b=11 cm$

c = 6 c m

$c=6 cm$

$\cos\color{#007DDC}{A}$	$=$ $=$	$\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

$\cos\color{#007DDC}{\theta}$	$=$ $=$	$\frac{\color{#00880A}{11}^2+\color{#9a00c7}{6}^2-\color{#007DDC}{15}^2}{2(\color{#00880A}{11})(\color{#9a00c7}{6})}$	Substitute the values

$cos theta$	$=$	$(121+36-225)/(132)$	Simplify

$cos theta$	$=$	$-68/132$

$cos theta$	$=$	$-0.515151…$
$theta$	$=$	$cos^(-1) -0.515151515$	Get the inverse of the cosine

Simplify this further by evaluating

$cos^(-1) -0.515151515$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$-0.515151515$

$4.$ Press $=$

The result will be:

$121.00758°$

Proceed with solving for

$theta$ .

$cos^(-1) -0.515151515=121.00758°$

$theta$	$=$	$cos^(-1) -0.515151515$
$theta$	$=$	$121.00758°$
$theta$	$=$	$121°0’27”$	Press DMS on the calculator
$theta$ or $A$	$=$	$121°$	Round off to the nearest degree

$121°$

Question 2 of 5

Find

$theta$

Round your answer to the nearest degree

$theta=$ (83) $°$

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Cosine Law

$\color{#007DDC}{a}^2=\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-2\color{#00880A}{b}\color{#9a00c7}{c}\cos\color{#007DDC}{A}$

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 Cosine Law (for non-right angled triangles)

a) Given 3 sides to find an angle

or

b) Given 2 sides and 1 angle 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

Since

$3$ sides are given, use the Cosine Law.

First, label the triangle according to the Cosine Law.

Substitute the three known values to the Cosine Law to find

$theta$ or $A$ .

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

$a, b$ and

$c$ .

$A=theta$

$a=10 cm$

$b=8 cm$

$c=7 cm$

$\cos\color{#007DDC}{A}$	$=$	$\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

$\cos\color{#007DDC}{\theta}$	$=$	$\frac{\color{#00880A}{8}^2+\color{#9a00c7}{7}^2-\color{#007DDC}{10}^2}{2(\color{#00880A}{8})(\color{#9a00c7}{7})}$	Substitute the values

$cos theta$	$=$	$(64+49-100)/(112)$	Simplify

$cos theta$	$=$	$13/112$

$cos theta$	$=$	$0.11607143$
$theta$	$=$	$cos^(-1) 0.11607143$	Get the inverse of the cosine

Simplify this further by evaluating

$cos^(-1) 0.11607143$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$0.11607143$

$4.$ Press $=$

The result will be:

$83.33457°$

Proceed with solving for

$theta$ .

$cos^(-1) 0.11607143=83.33457°$

$theta$	$=$	$cos^(-1) 0.11607143$
$theta$	$=$	$83.33457°$
$theta$	$=$	$83°20’$	Press DMS on the calculator
$theta$ or $A$	$=$	$83°$	Round off to the nearest degree

$83°$

Question 3 of 5

Find

$theta$

Round your answer to the nearest degree

$theta=$ (93) $°$

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Cosine Law

$\color{#007DDC}{a}^2=\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-2\color{#00880A}{b}\color{#9a00c7}{c}\cos\color{#007DDC}{A}$

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 Cosine Law (for non-right angled triangles)

a) Given 3 sides to find an angle

or

b) Given 2 sides and 1 angle 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

Since

$3$ sides are given, use the Cosine Law.

First, label the triangle according to the Cosine Law.

Substitute the three known values to the Cosine Law to find

$theta$ or $A$ .

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

$a, b$ and

$c$ .

$A=theta$

$a=8 m$

$b=5 m$

$c=6 m$

$\cos\color{#007DDC}{A}$	$=$	$\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

$\cos\color{#007DDC}{\theta}$	$=$	$\frac{\color{#00880A}{5}^2+\color{#9a00c7}{6}^2-\color{#007DDC}{8}^2}{2(\color{#00880A}{5})(\color{#9a00c7}{6})}$	Substitute the values

$cos theta$	$=$	$(25+36-64)/(60)$	Simplify

$cos theta$	$=$	$(-3)/60$

$cos theta$	$=$	$-0.05$
$theta$	$=$	$cos^(-1) -0.05$	Get the inverse of the cosine

Simplify this further by evaluating

$cos^(-1) -0.05$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$-0.05$

$4.$ Press $=$

The result will be:

$92.86598°$

Proceed with solving for

$theta$ .

$cos^(-1) -0.05=92.86598°$

$theta$	$=$	$cos^(-1) -0.05$
$theta$	$=$	$92.86598°$
$theta$	$=$	$92°51’$	Press DMS on the calculator
$theta$ or $A$	$=$	$93°$	Round off to the nearest degree

$93°$

Question 4 of 5

Noah (

$A$ ) is ready to shoot for a goal. When he is

$6.8$ m from one post,

$8.1$ m from the other post and the goal mouth is

$7.3$ m wide, what is the size of the angle

$(theta)$ for which Noah is to score a goal to the nearest minute?

$theta=$ (57) $°$ (53) $'$

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Cosine Law

$\color{#007DDC}{a}^2=\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-2\color{#00880A}{b}\color{#9a00c7}{c}\cos\color{#007DDC}{A}$

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 Cosine Law (for non-right angled triangles)

a) Given 3 sides to find an angle

or

b) Given 2 sides and 1 angle 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

Since the scenario forms a triangle where

$3$ sides are given, use the Cosine Law.

First, label the triangle according to the Cosine Law.

Substitute the three known values to the Cosine Law to find $A$ .

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

$a, b$ and

$c$ .

$a=7.3$ m

$b=8.1$ m

$c=6.8$ m

$\cos\color{#007DDC}{A}$	$=$	$\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

$\cos\color{#007DDC}{A}$	$=$	$\frac{\color{#00880A}{8.1}^2+\color{#9a00c7}{6.8}^2-\color{#007DDC}{7.3}^2}{2(\color{#00880A}{8.1})(\color{#9a00c7}{6.8})}$	Substitute the values

$cosA$	$=$	$(65.61+46.24-53.29)/(110.16)$	Simplify

$cosA$	$=$	$58.56/110.16$

$cosA$	$=$	$0.53159$
$A$	$=$	$cos^(-1) 0.53159$	Get the inverse of the cosine

Simplify this further by evaluating

$cos^(-1) 0.53159$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$0.53159$

$4.$ Press $=$

The result will be:

$57.887°$

Proceed with solving for

$A$ .

$cos^(-1) 0.53159=57.887°$

$A$	$=$	$cos^(-1) 0.53159$
$A$	$=$	$57.887°$
$A$	$=$	$57°53’13”$	Press DMS on the calculator
$A$	$=$	$57°53’$	Round off to the nearest minute

$57°53’$

Question 5 of 5

Find

$theta$

Round your answer to the nearest minute

$theta=$ (33) $°$ (50) $'$

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Cosine Law

$\color{#007DDC}{a}^2=\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-2\color{#00880A}{b}\color{#9a00c7}{c}\cos\color{#007DDC}{A}$

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 Cosine Law (for non-right angled triangles)

a) Given 3 sides to find an angle

or

b) Given 2 sides and 1 angle 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

Since

$3$ sides are given, use the Cosine Law.

First, label the triangle according to the Cosine Law.

Rewrite the Cosine Law according to which angle is missing, then substitute the three known values to find

$theta$ or $B$ .

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

$a, b$ and

$c$ .

$B=theta$

$a=12 cm$

$b=9 cm$

$c=16 cm$

$\cos\color{#007DDC}{A}$	$=$	$\frac{\color{#00880A}{b}^2+\color{#9a00c7}{c}^2-\color{#007DDC}{a}^2}{2\color{#00880A}{b}\color{#9a00c7}{c}}$

$\cos\color{#00880A}{B}$	$=$	$\frac{\color{#007DDC}{a}^2+\color{#9a00c7}{c}^2-\color{#00880A}{b}^2}{2\color{#007DDC}{a}\color{#9a00c7}{c}}$	Rewrite the Cosine Law

$\cos\color{#00880A}{\theta}$	$=$	$\frac{\color{#007DDC}{12}^2+\color{#9a00c7}{16}^2-\color{#00880A}{9}^2}{2\color{#007DDC}{(12)}\color{#9a00c7}{(16)}}$	Substitute the values

$cos theta$	$=$	$(144+256-81)/(384)$	Simplify

$cos theta$	$=$	$319/384$

$theta$	$=$	$cos^(-1) (319/384)$	Get the inverse of the cosine

$theta$	$=$	$cos^(-1) 0.830729$

Simplify this further by evaluating

$cos^(-1) 0.830729$ using the calculator:

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

$2.$ Press $cos$

$3.$ Press

$0.830729$

$4.$ Press $=$

The result will be:

$33.8263°$

Proceed with solving for

$theta$ .

$cos^(-1) 0.830729=33.8263°$

$theta$	$=$	$cos^(-1) 0.830729$
$theta$	$=$	$33.8263°$
$theta$	$=$	$33°49’34”$	Press DMS on the calculator
$theta$ or $B$	$=$	$33°50’$	Round off to the nearest minute

$33°50’$

Cosine Rule: Solving for an Angle

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Quiz summary

Information

1. Question

Hint

Excellent!

Cosine Rule

When to use the Cosine Rule (for non-right angled triangles)

Calculator Buttons to Use

2. Question

Hint

Well Done!

Cosine Law

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

Calculator Buttons to Use

3. Question

Hint

Fantastic!

Cosine Law

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

Calculator Buttons to Use

4. Question

Hint

Good Job!

Cosine Law

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

Calculator Buttons to Use

5. Question

Hint

Exceptional!

Cosine Law

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

Calculator Buttons to Use

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