Maths - Sine Rule | Olly Britton

The two forms of the sine rule

What is the sine-on-top form of the sine rule?

\[\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\]

What is the sine-on-bottom form of the sine rule?

\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]

Where is angle $A$ in relation to the side $a$?

Opposite.

Where is the side $c$ in relation to the angle $C$?

Opposite.

How do you draw something like “quadrilateral $ABCD$”?

Draw the quadrilateral and label the sides moving clockwise.

Why can you sometimes draw two different triangles when using the sine rule?

Because $\sin(\theta) = \sin(180 - \theta)$.

What relationship does this graph represent?

\[\sin(\theta) = \sin(180 - \theta)\]

How do you start the proof of the sine rule?

Draw a vertical line $h$ that goes from one vertex of the triangle and intersects another at $90^{\circ}$.

If $\sin(A) = \frac{h}{b}$ and $\sin(B) = \frac{h}{a}$, then how could you turn it into the sine rule?

\[h = b\sin(A)\] \[\sin(B) = \frac{(b\sin(A))}{a}\] \[\frac{\sin(B)}{b} = \frac{\sin(A)}{a}\]

How could you write $sin(A)$ in terms of $b$ and $h$?

\[\sin(A) = \frac{h}{b}\]

How could you write $sin(B)$ in terms of $a$ and $h$?

\[\sin(B) = \frac{h}{a}\]

If the sine rule has two solutions, then the two angles will be what?