Sine, Cosine Rule & Area of Triangles Flashcards

All 7 cards in this deck

What is the sine rule?

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ — use it when you know a side together with the angle opposite it.

True or false? Using the sine rule to find an angle always gives just one possible answer.

False. There is also an obtuse possibility, $180° - \theta$, which must be checked (the ambiguous case).

What is the cosine rule, in both its forms?

$a^2 = b^2 + c^2 - 2bc\cos A$ for a side; rearranged for an angle, $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.

What is the formula for the area of a triangle given two sides and the angle between them?

Area $= \frac{1}{2}ab\sin C$, where $C$ is the included angle between sides $a$ and $b$.

What are the steps to find a missing side from the area of a triangle?

e.g. area 24 cm², one side 12 cm, included angle 30°

1. Substitute into Area $= \frac{1}{2}ab\sin C$: $24 = \frac{1}{2} \times 12 \times b \times \sin 30°$.
2. Simplify the right-hand side: $24 = 3b$.
3. Divide: $b = 8$ cm.

When do you use the cosine rule rather than the sine rule?

When you know all 3 sides (to find an angle), or 2 sides and the included angle (to find the third side); otherwise use the sine rule.

True or false? In the formula $\text{area} = \frac{1}{2}ab\sin C$, $C$ can be any of the three angles of the triangle.

False. $C$ must be the angle between (included by) the two sides $a$ and $b$.