Sine Rule: Solving for an Angle
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Question 1 of 5
1. Question
Find `theta`
Round your answer to the nearest minute- `theta=` (67)`°` (32)`'`
Hint
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Nice Job!
Incorrect
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 angleorb) Given 2 angles 1 side 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 functionDegree-to-Minute Conversion
`1` degree `=60` minutesMinute-to-Second Conversion
`1` minute `=60` secondsRounding 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
2. Question
Find `theta`
Round your answer to the nearest degree- `theta=` (62)`°`
Hint
Help VideoCorrect
Keep Going!
Incorrect
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 angleorb) Given 2 angles 1 side 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 functionDegree-to-Minute Conversion
`1` degree `=60` minutesMinute-to-Second Conversion
`1` minute `=60` secondsRounding 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
3. Question
Find `theta`
Round your answer to the nearest degree- `theta=` (74)`°`
Hint
Help VideoCorrect
Excellent!
Incorrect
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 angleorb) Given 2 angles 1 side 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 functionDegree-to-Minute Conversion
`1` degree `=60` minutesMinute-to-Second Conversion
`1` minute `=60` secondsRounding 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
4. Question
Find the obtuse angle `alpha`
Round your answer to the nearest degree- `alpha=` (131)`°`
Hint
Help VideoCorrect
Excellent!
Incorrect
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 angleorb) Given 2 angles 1 side 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 functionDegree-to-Minute Conversion
`1` degree `=60` minutesMinute-to-Second Conversion
`1` minute `=60` secondsRounding 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
5. Question
Find the obtuse angle `alpha`
Round your answer to the nearest degree- `alpha=` (150)`°`
Hint
Help VideoCorrect
Good Job!
Incorrect
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 angleorb) Given 2 angles 1 side 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 functionDegree-to-Minute Conversion
`1` degree `=60` minutesMinute-to-Second Conversion
`1` minute `=60` secondsRounding 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°`
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