GCSE(F), GCSE(H),

**Trigonometric Ratios** are the relationship between an angle and the three sides of a right angled triangle.

The ratio of the *opposite* side to the *hypotenuse* is the **sine** of the angle, shortened to **sin**. The opposite side is the side opposite the angle.

`sin(x) = frac(text(Opposite))(text(Hypotenuse))`

If an angle and one length (hypotenuse or opposite) are known, find the sin value for that angle using a calculator (or tables).

If the two lengths (hypotenuse and opposite) are known, the formula gives the sin of the angle. Using a calculator, look up the value of the angle using the inverse operation (which is written as sin^{-1}).

1. What is the length of AB?

Answer: 7.053 cm

sin *x* = `frac(text(opposite))(text(hypotenuse))`

sin 36 = `frac(text(AB))(12)`

0.5878 x 12 = AB, therefore AB = 7.053

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

Answer: 34.2º

sin *x* = `frac(text(opposite))(text(hypotenuse))`

sin *x* = `frac(9)(16)`

sin *x* = 0.5625

*x* = sin^{-1} 0.5625; therefore *x* = 34.23º

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