Law of Cosines: Solving for an Angle

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Question 1 of 5

1. Question

Find `theta`

Round your answer to the nearest degree

`theta=` (121)`°`

Hint

Help Video

Correct

Excellent!

Incorrect

Cosine Rule

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

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 Rule (for non-right angled triangles)

a) Given 3 sides to find an angle

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 Rule.

First, label the triangle according to the Cosine Rule.

Substitute the three known values to the Cosine Rule 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=15 cm`

`b=11 cm`

`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

2. Question

Find `theta`

Round your answer to the nearest degree

`theta=` (83)`°`

Hint

Help Video

Correct

Well Done!

Incorrect

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

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

3. Question

Find `theta`

Round your answer to the nearest degree

`theta=` (93)`°`

Hint

Help Video

Correct

Fantastic!

Incorrect

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

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

4. Question

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)`'`

Hint

Help Video

Correct

Good Job!

Incorrect

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

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

5. Question

Find `theta`

Round your answer to the nearest minute

`theta=` (33)`°` (50)`'`

Hint

Help Video

Correct

Exceptional!

Incorrect

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

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’`

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