C2 AS maths

Sine and Cosine Formulae

The Sine Rule

The sine rule is an important rule relating the sides and angles of any triangle (it doesn't have to be right-angled!):

If a, b and c are the lengths of the sides opposite the angles A, B and C in a triangle, then:

a = b = c

sinA sinB sinC

If you wanted to find an angle, you can write this as:

sinA = sinB = sinC

a b c

The Cosine Rule

This also works in any triangle:

c2 = a2 + b2 - 2abcosC

which can also be written as:

a2 = b2+ c2 - 2bccosA

The area of a triangle

The area of any triangle is ½absinC (using the above notation).

This formula is useful if you don't know the height of a triangle (since you need to know the height for ½ base × height).

Radians

Radians, like degrees, are a way of measuring angles.

One radian is equal to the angle formed when the arc opposite the angle is equal to the radius of the circle. So in the above diagram, the angle  is equal to one radian since the arc AB is the same length as the radius of the circle.

Now, the circumference of the circle is 2r, where r is the radius of the circle. So the circumference of a circle is 2 larger than its radius. This means that in any circle, there are 2 radians.

Therefore 360° = 2 radians.

Therefore 180° = radians.

So one radian = 180/ degrees and one degree = /180 radians.

Therefore to convert a certain number of degrees in to radians, multiply the number of degrees by /180 (for example, 90° = 90 ×/180 radians = /2). To convert a certain number of radians into degrees, multiply the number of radians by 180/ .

Arc Length

The length of an arc of a circle is equal to r where  is the angle, in radians, subtended by the arc at the centre of the circle (see below diagram if you don't understand). So in the below diagram, s = r

Area of Sector

The area of a sector of a circle is ½r2 where r is the radius and  is the angle in radians subtended by the arc at the centre of the circle. So in the below diagram, the shaded area is equal to ½ r2 

Sin, Cos and Tan

The Sine, Cosine and Tangents of Common Angles

30 (/6) 45 (/4) 60 (/3)

sin 1 / 2 1 /2 3 / 2

cos 3/ 2 1 / 2 1 / 2

tan 1 / 3 1 3

These occur frequently and should be remembered.