GCSE(H),

The sine rule can be used for any triangle, whether right angles or not. Note that the letters relating to the side are opposite the corresponding letters for the angles, i.e. a and A; b and B and c and C.

The rule is:

`frac(a)(sin A) = frac(b)(sin B) = frac(c)(sin C)`

Use the sine rule when any two sides and an angle, or two angles and a side are known, the sine rule can be used.

The sine rule can also be used with the angles on top: `frac(sin A)(a) = frac(sin B)(b) = frac(sin C)(c)`

1. What is the value of the angle *x*?

Answer: 50.92ยบ

Using the sine rule:

`frac(sin A)(a) = frac(sin B)(b)`

`frac(sinA)(12) = frac(sin 76)(15)`

`sin A = frac(12 xx sin 76)(15)`

`sin A = 0.776`

`A = sin^-1 0.776`

`A = 50.917`

2. How long is the side *x*?

Answer: 10.2 cm

The unknown length is opposite an unknown angle.

The unknown angle = 180 - 77 - 43 = 60.

`frac(x)(sin 60) = frac(8)(sin 43)`

`x = frac(sin 60 xx 8)(sin 43)`

`x = 10.158`

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