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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 .
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