# Maths - Sine Rule

> Source: https://ollybritton.com/notes/a-level/maths/topics/sine-rule/ · Updated: 2020-11-18 · Tags: maths, year-1, school

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

##### ![PHOTO](sine-same-inverse.png) 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??
$$
h = b\sin(A)
$$

$$
\sin(B) = \frac{(b\sin(A))}{a}
$$

$$
\frac{\sin(B)}{b} = \frac{\sin(A)}{a}
$$

##### ![PHOTO](sine-rule-proof.png) How could you write $sin(A)$ in terms of $b$ and $h$??
$$
\sin(A) = \frac{h}{b}
$$

##### ![PHOTO](sine-rule-proof.png) How could you write $sin(B)$ in terms of $a$ and $h$??
$$
\sin(A) = \frac{h}{a}
$$

##### If the sine rule has two solutions, then the two angles will be what??
* Obtuse
* Acute

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