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Double angle formulae

The following are important trigonometric relationships (it is unlikely that you will need to know how to prove them and they may be given in your formula book- check!):

sin(A + B) = sinAcosB + cosAsinB
cos(A + B) = cosAcosB - sinAsinB
tan(A + B) =   tanA + tanB
                     1 - tanAtanB

To find sin(A - B), cos(A - B) and tan(A - B), just change the + signs in the above identities to -

sin(A - B) = sinAcosB - cosAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A - B) =   tanA - tanB
                    1 + tanAtanB

Double Angle Formulae
sin(A + B) = sinAcosB + cosAsinB
Replacing B by A in the above formula becomes:
sin(2A) = sinAcosA + cosAsinA
so sin2A = 2sinAcosA

similarly, cos2A = cos²A - sin²A
Replacing cos²A by 1 - sin²A (see Pythagorean identities) in the above formula gives:
cos2A = 1 - 2sin²A
Replacing sin²A by 1 - cos²A gives:
cos2A = 2cos²A - 1

It can also be shown that:
tan2A =    2tanA  
              1 - tan²A

Pythagorean Identities

This important identity can be derived as a direct result of Pythagoras's theorem, when applied to angles in trigonometry:

sin²x + cos²x = 1         (1)

By dividing each of these terms by sin²x, we can derive a second identity:

1 + cot²x = cosec²x

By dividing (1) by cos²x, we arrive at the third (and final) identity:

tan²x + 1 = sec²x


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 ø 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 2pr, where r is the radius of the circle. So the circumference of a circle is 2p larger than its radius. This means that in any circle, there are 2p radians.
Therefore 360º = 2p radians.
Therefore 180º = p radians.
So one radian = 180/p degrees and one degree = p/180 radians.

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

Arc Length
The length of an arc of a circle is equal to rø, where ø is the angle, in radians, subtended by the arc at the centre of the circle. So in the below diagram, s = rø .

Area of Sector
The area of a sector of a circle is ½ r² ø, where r is the radius and ø 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 ø .

Sec, cosec, cot

Secant, cosecant and cotangent
secant x or sec x  =     1    
                                cos x
cosecant x, or cosec x =     1  
                                      sin x
cotangent x, or cot x =   1  
                                   tan x
Note, sec x is not the same as arccos x.

If sec x = 2, cos x = ½, x = 60º

Sine and Cosine Formulae

Sine and Cosine Formulae
sin x = sin (180 - x)
cos x = -cos (180 - x)

The Sine Rule
This works in any triangle:

   a    =    b     = c  
  sinA     sinB      sinC

alternatively, sinA = sinB = sinC
                    a         b         c

NOTE: the triangle is labelled as follows:

The Cosine Rule
c² = a² + b² - 2abcosC
/ which can also be written as:
a² = b² + c² - 2bccosA

This also works in any triangle.  

The area of a triangle
The area of any triangle is ½ absinC (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).

Solving Trigonometric Equations

The various trigonometric formulae and identities can be used to help solve trigonometric equations. Here is a summary of the most important trigonometric formulae you should know:
sin²x + cos²x = 1
1 + cot²x = cosec²x
tan²x + 1 = sec²x
cos2x = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x
sin2x = 2sinx cosx
tanx = sinx

Solve 2cos²x + 3sinx = 3, giving your answer in radians for 0< x <p.
\ 2cos²x + 3sinx - 3 = 0
We need to get everything in terms of sinx or everything in terms of cosx. Since we know that cos²x = 1 - sin²x:
\ 2(1 - sin²x) + 3sinx - 3 = 0
\ 2 - 2sin²x + 3sinx - 3 = 0
\ -2sin²x + 3sinx - 1 = 0
\ 2sin²x - 3sinx + 1 = 0
\ (2sinx - 1)(sinx - 1) = 0
\ sin x = ½ or sin x = 1
x = p/6, 5p/6, p/2

Remember, if sinx = 1,  x = p/2, 5p/2, 9p/2, ... and the same is true for arcsin(½). In the question, you are asked for values of x between 0 and p. You must write down all of the appropriate solutions.

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