Cosine Rule

Firstly, we need appropriately label the sides of this triangle. Firstly, we set a=x, and therefore we get that A=19, since it is the angle opposite. It doesn’t matter how we label the other two sides, so here we’ll let b=86 cm and c=65 cm. Now, subbing these values into the cosine rule equation, we get

 

x^2=86^2+65^2-(2\times86\times65\times\cos(19\degree))=7,396+4,225-11,180\cos(19\degree)

 

Then, taking the square root, and putting it into the calculator, we get,

 

x=\sqrt{7,396+4,225-11,180\cos(19\degree)}=32 cm (2 sf).

As always, we must label our triangle. Firstly, assign the thing we’re looking for to be A=x, and therefore make the side opposite to it a=6 mm. Then, it doesn’t matter how we choose the other two sides, so we will let b=5 mm and c=7 mm.

 

Here, we will use the rearranged version of the formula that looks like

 

\cos A=\dfrac{b^2+c^2-a^2}{2bc},

 

So, subbing these values into the equation, we get

 

\begin{aligned} \cos x &=\dfrac{5^2+7^2-6^2}{2\times5\times7} \\ \\ &=\dfrac{25+49-36}{70}\end{aligned}

 

Taking \cos^{-1} of both sides, and putting it into a calculator, we get

 

x=\cos^{-1}\left(\dfrac{25+49-36}{70}\right)=57.1\degree (3 sf).

Firstly, we need appropriately label the sides of this triangle. Firstly, we set a=x, and therefore we get that A=40\degree, since it is the angle opposite. It doesn’t matter how we label the other two sides, so here we’ll let b=6 cm and c=4.5 cm. Now, substiuting  these values into the cosine rule equation, we get,

 

x^2=6^2+4.5^2-(2\times6\times4.5\times\cos(40\degree))=36+20.25-54\cos(40\degree)

 

Then, taking the square root, and putting it into the calculator, we get,

 

x=\sqrt{56.25-54\cos(40\degree)}=3.9 cm (2 sf).

We can use the rearranged version of the formula that looks like,

 

\cos A=\dfrac{b^2+c^2-a^2}{2bc}

 

So, subbing these values into the equation, we get

 

\begin{aligned} \cos x &=\dfrac{7^2+10^2-12^2}{2\times7\times10} \\ \\ &=\dfrac{49+100-144}{140} \end{aligned}

 

Taking \cos^{-1} of both sides, and putting it into a calculator, we get

 

x=\cos^{-1}\left(\dfrac{5}{140}\right)=88.0\degree (1 dp).

Substituting values into the cosine rule equation, we get,

 

x^2=5.2^2+3.5^2-(2\times5.2\times3.5\times\cos(28\degree))=27.04+12.25-36.4\cos(28\degree)

 

Then, taking the square root, and putting it into the calculator, we get,

 

x=\sqrt{39.29-36.4\cos(28\degree)}=2.7 cm (2 sf).