## 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.

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## 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.

**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]**

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

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]**

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.

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.

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)

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### 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).

### Worksheets and Exam Questions

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