Sine Rule
Sine Rule Revision
The Sine Rule
When we first learn the sine function, we learn how to use it to find missing side-lengths & angles in right-angled triangles. The sine rule is an equation that can help us find missing side-lengths and angles in any triangle.
Make sure you are happy with the following topics before continuing:
![The Sine Rule Formula](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Formula.png)
The Sine Rule Formula
Looking at the triangle below, the sine rule is:
\dfrac{\textcolor{limegreen}{a}}{\sin \textcolor{limegreen}{A}}=\dfrac{\textcolor{blue}{b}}{\sin \textcolor{blue}{B}}=\dfrac{\textcolor{red}{c}}{\sin \textcolor{red}{C}}
In this topic, we’ll go through examples of how to use the sine rule to find missing angles and missing sides.
![The Sine Rule Formula](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Formula.png)
Example 1: Sine Rule to Find a Length
Use the sine rule to find the side-length marked x to 3 s.f.
[2 marks]
![Sine rule to find a length](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Example1.png)
First we need to match up the letters in the formula with the sides we want, here:
a=x, A=21\degree, b = 23 and B = 35\degree
![Sine rule to find a length](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Example2.png)
Next we’re ready to substitute the values into the formula. Doing so gives us:
\dfrac{x}{\sin(21°)}=\dfrac{23}{\sin(35°)}
Multiplying both sides by \sin(21°):
x=\dfrac{23}{\sin(35°)}\times\sin(21°)
Putting this into a calculator we get:
x=14.37029543...
x=14.4 (3 sf)
As in previous topics, there is no need to evaluate the sine functions until the final step.
Example 2: Sine Rule to Find an Angle
Use the sine rule to find the obtuse angle marked x to 2 s.f.
[2 marks]
![Sine rule to find an angle](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Example3.png)
As we have been asked to find a missing angle, we can use another version of the sine rule:
\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}
A=x, a=43, B=33\degree, b=25.
![Sine rule to find an angle](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Example4.png)
Substituting these values into the formula, we get:
\dfrac{\sin x}{43}=\dfrac{\sin(33°)}{25}
Multiply both sides by 43 to get:
\sin x=\dfrac{43\sin(33°)}{25}
Then, taking \sin^{-1} of both sides, we get:
x= \sin^{-1}\bigg(\dfrac{43\sin(33°)}{25}\bigg)
x=69.5175049...°
However, the question asked for an obtuse angle, but we got an acute answer – why?
It’s because we can draw two different (but both correct) triangles using the information we were given at the start.
![Sine rule to find an angle](https://mmerevise.co.uk/app/uploads/2020/09/The-Sine-Rule-Example5.png)
This is the ambiguous case of the sine rule, and it occurs when you have 2 sides and an angle that doesn’t lie between them.
To find the obtuse angle, simply subtract the acute angle from 180:
180-69.5175049=110.4824951
x=110\degree (2 sf)
Sine Rule Example Questions
Question 1: Use the sine rule to find the side-length marked x in the below triangle to 3 significant figures.
[3 marks]
First, we need to find the angle opposite to the missing side as it is not given in the question. Using all the angles in a triangle add to 180 degrees we get that:
A=180\degree-40\degree-94\degree=46\degree
Now we have enough information to properly label the triangle and substitute values into the sine rule:
\dfrac{x}{\sin(46\degree)}=\dfrac{10.5}{\sin(94\degree)}
Solving for x we get:
x=\dfrac{10.5}{\sin(94\degree)}\times\sin(46\degree)=7.571511726...
x=7.57 (3 sf).
Question 2: Use the sine rule to find the side-length marked x in the below triangle to 3 significant figures.
[2 marks]
Here we are able to use the sine rule straightaway:
\dfrac{x}{\sin(30\degree)}=\dfrac{5}{\sin(80\degree)}
Multiplying both sides of the equation by \sin(30\degree):
x=\dfrac{5}{\sin(80\degree)}\times\sin(30\degree)=2.538566...
x=2.54 cm (3 sf).
Question 3: Use the sine rule to find the obtuse angle x on the diagram below to 3 significant figures.
[3 marks]
Here we are able to use the sine rule straightaway:
\dfrac{\sin(x\degree)}{12}=\dfrac{\sin(15\degree)}{7}
Multiplying both sides of the equation by 12 we find:
\sin(x)=\dfrac{12\times\sin(15\degree)}{7}=0.4436897916
Taking the inverse sine of both sides:
x=\sin^{-1}(0.4436897916)=26.33954244\degree
However considering the diagram, the angle is clearly obtuse (greater than 90 degrees). This is the ambiguous case of the sine rule and it occurs when you have 2 sides and an angle that doesn’t lie between them. To find the obtuse angle, simply subtract the acute angle from 180:
180\degree-26.33954244\degree =153.6604576
=154\degree (3 sf).
Instead of typing the full number into the calculator for each step of the calculation, you can use the ANS button to save time.
Question 4: Use the sine rule to find the angle CAB on the diagram below to 3 significant figures.
[2 marks]
We are able to use the sine rule straightaway:
\dfrac{\sin(x\degree)}{6.5}=\dfrac{\sin(52\degree)}{12}
Multiplying both sides of the equation by 6.5 we find that:
\sin(x)=6.5 \times \dfrac{\sin(52\degree)}{12} =0.4268391582
Taking the inverse sine of both sides and keeping the answer from the previous step on our calculator, we get:
x=\sin^{-1}(ANS)=25.26713177
x=25.3 \degree (3 sf).
Question 5: Use the sine rule to find the side-length marked x below to 3 significant figures.
[2 marks]
Applying the sine rule:
\dfrac{x}{\sin(35\degree)}=\dfrac{6}{\sin(68\degree)}
Multiplying both sides of the equation by \sin(35\degree), we find:
x=\dfrac{6}{\sin(68\degree)}\times\sin(35\degree)=3.711732685...
x=3.71 cm (3 sf).
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