Cosine Rule: Solving for an Angle

Years

Try VividMath Premium to unlock full access

Start Free Trial

Time limit: 0

Quiz summary

0 of 5 questions completed

Questions:

Information

–

You have already completed the quiz before. Hence you can not start it again.

Quiz is loading...

You must sign in or sign up to start the quiz.

You have to finish following quiz, to start this quiz:

Lets see your results!

Points

Score

Time

Retry Quiz

Answered
Review

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’`

Quizzes

Intro to Trigonometric Ratios (SOH CAH TOA) 1
Intro to Trigonometric Ratios (SOH CAH TOA) 2
Round Angles (Degrees, Minutes, Seconds)
Evaluate Trig Expressions using a Calculator 1
Evaluate Trig Expressions using a Calculator 2
Trig Ratios: Solving for a Side 1
Trig Ratios: Solving for a Side 2
Trig Ratios: Solving for an Angle
Angles of Elevation and Depression
Trig Ratios Word Problems: Solving for a Side
Trig Ratios Word Problems: Solving for an Angle
Area of Non-Right Angled Triangles 1
Area of Non-Right Angled Triangles 2
Sine Rule: Solving for a Side
Sine Rule: Solving for an Angle
Cosine Rule: Solving for a Side
Cosine Rule: Solving for an Angle
Trigonometry Word Problems 1
Trigonometry Word Problems 2
Trigonometry Mixed Review: Part 1 (1)
Trigonometry Mixed Review: Part 1 (2)
Trigonometry Mixed Review: Part 1 (3)
Trigonometry Mixed Review: Part 1 (4)
Trigonometry Mixed Review: Part 2 (1)
Trigonometry Mixed Review: Part 2 (2)
Trigonometry Mixed Review: Part 2 (3)