The Cosine Rule Flashcards

All 8 cards in this deck

State the cosine rule for finding a missing side.

$a^2=b^2+c^2-2bc\cos A$

$A$ is the angle between sides $b$ and $c$, opposite side $a$.

State the cosine rule rearranged to find a missing angle.

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

What are the steps to find a side using the cosine rule?

e.g. sides $8$ cm and $11$ cm with an included angle of $72^\circ$

1. Substitute: $a^2=8^2+11^2-2\times 8\times 11\times\cos 72^\circ$
2. Correct order of operations: $64+121-54.4=130.6$
3. Square root: $a=11.4$ cm (3 s.f.)

What are the steps to find an angle using the cosine rule?

e.g. find the angle opposite the $7$ cm side in a triangle with sides $5$ cm, $8$ cm, $7$ cm

1. Substitute into $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$: $\cos A=\dfrac{5^2+8^2-7^2}{2\times5\times8}$
2. Simplify: $\cos A=\dfrac{40}{80}=0.5$
3. Inverse cosine: $A=60^\circ$

True or false? In $a^2=b^2+c^2-2bc\cos A$ you may work out $b^2+c^2-2bc$ first and then multiply by $\cos A$.

False.

You must multiply $2bc$ by $\cos A$ first, then subtract that from $b^2+c^2$.

What are the steps to use the cosine rule in a bearings problem?

e.g. a ship sails $8$ km on a bearing of $060^\circ$, then $5$ km on a bearing of $150^\circ$; find its distance from the start

1. Sketch the journey with a north line at each turning point.
2. Use angle facts to find the interior angle of the triangle: $90^\circ$
3. Apply the cosine rule: $d^2=8^2+5^2-2\times8\times5\cos 90^\circ$, so $d=9.43$ km (3 s.f.)

True or false? If the cosine rule gives a negative value for $\cos A$, then angle $A$ is obtuse.

True. $\cos$ is negative for angles between $90^\circ$ and $180^\circ$, so $A$ is obtuse.

What are the steps to find the distance from the start of a two-leg journey given as bearings?

e.g. a boat sails $5$ km on a bearing of $050^\circ$, then $7$ km on a bearing of $110^\circ$

1. Sketch the journey and find the angle inside the triangle at the turning point: $180^\circ-(110^\circ-50^\circ)=120^\circ$
2. Two sides and the included angle, so use the cosine rule: $d^2=5^2+7^2-2\times5\times7\times\cos120^\circ=109$
3. Square root: $d\approx10.4$ km