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Law of Cosines: Solving for an AngleLaw 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 VideoCorrect
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 angleorb) Given 2 sides and 1 angle to find the other sideCalculator Buttons to Use
`sin` `=` Sine function`cos` `=` Cosine function`tan` `=` Tangent functionDMS or `° ‘ ‘ ‘` `=` Degree/Minute/SecondShift or 2nd F or INV `=` Inverse function`=` `=` Equal functionSince `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 VideoCorrect
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 angleorb) Given 2 sides and 1 angle to find the other sideCalculator Buttons to Use
`sin` `=` Sine function`cos` `=` Cosine function`tan` `=` Tangent functionDMS or `° ‘ ‘ ‘` `=` Degree/Minute/SecondShift or 2nd F or INV `=` Inverse function`=` `=` Equal functionSince `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 VideoCorrect
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 angleorb) Given 2 sides and 1 angle to find the other sideCalculator Buttons to Use
`sin` `=` Sine function`cos` `=` Cosine function`tan` `=` Tangent functionDMS or `° ‘ ‘ ‘` `=` Degree/Minute/SecondShift or 2nd F or INV `=` Inverse function`=` `=` Equal functionSince `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 VideoCorrect
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 angleorb) Given 2 sides and 1 angle to find the other sideCalculator Buttons to Use
`sin` `=` Sine function`cos` `=` Cosine function`tan` `=` Tangent functionDMS or `° ‘ ‘ ‘` `=` Degree/Minute/SecondShift or 2nd F or INV `=` Inverse function`=` `=` Equal functionSince 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 VideoCorrect
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 angleorb) Given 2 sides and 1 angle to find the other sideCalculator Buttons to Use
`sin` `=` Sine function`cos` `=` Cosine function`tan` `=` Tangent functionDMS or `° ‘ ‘ ‘` `=` Degree/Minute/SecondShift or 2nd F or INV `=` Inverse function`=` `=` Equal functionSince `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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