Examples using cosine rule

Example 1

You can use the cosine rule (If you can remember it – but why try) here it is anyway:

What is the length B to C in the following diagram?

What is the length of B to C

Cosine rule is: $a^2=b^2+c^2-2bc\ cosA$

Redraw the diagram

Redraw the diagram to include cosine

Therefore $a^2=b^2+c^2-2bc\ cosA$

Becomes: $a^2=7^2+3^2-2times7times3cos35^@$

$a^2=49+8-42cos35^@$

$a^2=58-42cos35^@$

$a^2=58-34.4$

$a^2=23.6$

$a=sqrt 23.6$

$a=4.85cm$

Example 2

In the diagram below find $theta$

Find the angle in the diagram

Cosine rule is: $a^2=b^2+c^2-2bc\ cosA$

Redraw the diagram

Redraw the triangle to include cosine

Rearrange the cosine rule:

$-cosA=(a^2-b^2-c^2)/(2bc)$

$-cosA=(4^2-4.2^2-6.9^2)/(2times4.2times6.9)$

$-cosA=(16-17.64-47.61)/57.96$

$-cosA=(-49.25)/57.96$

Swap sides to cancel out the negative

$49.25/57.96=cosA$

$0.849=cosA$

Answer: $A=31.89^@$