Sine Rule – Formula, Revision and worksheets.

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 equation;

\dfrac{x}{\sin(46\degree)}=\dfrac{10.5}{\sin(94\degree)}

Multiplying both sides, and putting it into a calculator, we get

x=\dfrac{10.5}{\sin(94\degree)}\times\sin(46\degree)=7.57\text{ (3 s.f.)}

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), we find,

x=\dfrac{5}{\sin(80\degree)}\times\sin(30\degree)=2.54 \text{ cm (3 s.f.)}

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}\approx0.4437

Taking the inverse sine of both sides, 

x=\sin^{-1}(0.4437)=26.34\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-26.34\degree =154\degree \text{(3 s.f.)} 

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,

\sin(x)=6.5 \times \dfrac{\sin(52\degree)}{12}

Taking the inverse sine of both sides, 

x=\sin^{-1}(0.4268)=25.3 \degree \text{(3 s.f.)}

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)=3.71 \text{ cm (3 s.f.)}