Sine Rule — Finding a Side

Section 1 of 1

The Sine Rule — Finding a Side

Non-right-angled triangles

When there is no right angle, label each side with the small letter of the angle opposite it. Side $a$ faces angle $A$, and so on.

The Sine Rule

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$. Use it when you have an angle and its opposite side, plus one more angle.

In a triangle $A = 55^\circ$, $B = 65^\circ$ and $b = 5$. Find $a$.

$\dfrac{a}{\sin 55} = \dfrac{5}{\sin 65}$

$a = \dfrac{5\sin 55}{\sin 65} = 4.51 = 4.5$

$a = 4.5$

In a triangle $A = 35^\circ$, $B = 55^\circ$ and $b = 8$. Find $a$.

$\dfrac{a}{\sin 35} = \dfrac{8}{\sin 55}$

$a = \dfrac{8\sin 35}{\sin 55} = 5.60 = 5.6$

$a = 5.6$

In $\triangle ABC$, $|AB| = 5$, $|\angle BAC| = 45^\circ$ and $|\angle BCA| = 55^\circ$. Find $|BC|$.

$|BC| = a$ faces $A$; $|AB| = c = 5$ faces $C$

$\dfrac{a}{\sin 45} = \dfrac{5}{\sin 55} \Rightarrow a = \dfrac{5\sin 45}{\sin 55} = 4.31 = 4.3$

$|BC| = 4.3$

You try

In a triangle $A = 62^\circ$, $B = 41^\circ$ and $b = 12$. Find $a$.

$a = \dfrac{12\sin 62}{\sin 41}$.

$a = \dfrac{12\sin 62}{\sin 41} = 16.15 = 16.2$

$a = 16.2$

You try

In a triangle $A = 29^\circ$, $B = 73^\circ$ and $b = 12$. Find $a$.

$a = \dfrac{12\sin 29}{\sin 73}$.

$a = \dfrac{12\sin 29}{\sin 73} = 6.06 = 6.1$

$a = 6.1$

You try

A triangle has angles $47^\circ$ and $63^\circ$, and the side facing $47^\circ$ is $12$. Find $a$, the side facing $63^\circ$.

$\dfrac{a}{\sin 63} = \dfrac{12}{\sin 47}$.

$a = \dfrac{12\sin 63}{\sin 47} = 14.61 = 14.6$

$a = 14.6$

You try

A triangle has angles $47^\circ$ and $23^\circ$. The side facing $23^\circ$ is $8$. Find $x$, the side facing $47^\circ$.

$\dfrac{x}{\sin 47} = \dfrac{8}{\sin 23}$.

$x = \dfrac{8\sin 47}{\sin 23} = 14.97 = 15$

$x = 15$

The Sine Rule to find a side. Class 7: the Sine Rule to find an angle.