Cambridge International AS and A Level Mathematics Pure Mathematics 1

(Michael S) #1
Trigonometry

P1^


7


an equation: it is true when θ = 45° or 225°, but not when it takes any other value
in the range 0°  θ  360°.
By contrast, an identity is true for all values of the variable, for example

tan sin
cos
,tan sin
cos
30 30 ,tan(–
30

72 72

72

°= ° 33

°

°= °

°

99 399

399

°= °

°

) sin(–)
cos(–)

,

and so on for all values of the angle.
In this book, as in mathematics generally, we often use an equals sign where
it would be more correct to use an identity sign. The identity sign is kept for
situations where we really want to emphasise that the relationship is an identity
and not an equation.
Another useful identity can be found by applying Pythagoras’ theorem to any
point P(x, y) on the unit circle
y^2 + x^2 ≡ OP^2
(sin θ)^2 + (cos θ)^2 ≡ 1.
This is written as
sin^2 θ + cos^2 θ ≡ 1.

You can use the identities tan sin
cos
θ θ
θ
≡ and sin^2 θ + cos^2 θ ≡ 1 to prove other
identities are true.
There are two methods you can use to prove an identity; you can use either
method or a mixture of both.

Method 1
When both sides of the identity look equally complicated you can work with
both the left-hand side (LHS) and the right-hand side (RHS) and show that
LHS – RHS = 0.

ExAmPlE 7.3 Prove the identity cos^2 θ – sin^2 θ ≡ 2 cos^2 θ – 1.

SOlUTION
Both sides look equally complicated, so show LHS – RHS = 0.
So you need to show cos^2 θ – sin^2 θ – 2 cos^2 θ + 1 ≡ 0.
Simplifying:
cos^2 θ – sin^2 θ – 2 cos^2 θ + 1 ≡ – cos^2 θ – sin^2 θ + 1
≡ –(cos^2 θ + sin^2 θ) + 1

≡ –1 + 1 Using sin (^2) θ + cos (^2) θ = 1.
≡ 0 as required

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