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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**

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

**Introduction**

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 .

**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.

*Example*:

If sec x = 2, cos x = ½, x = 60º

**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).

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__

cosx

*Example*:

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, 5____p/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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