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Trigonometry III
Fundamental Trigonometric Identities.
By Mr Porter
Summary of Definitions
θ
Adjacent
Opposite
α
Opposite
Hypotenuse
cosecq =
Hypotenuse
Opposite
Adjacent
Hypotenuse
cosq =
secq =
Hypotenuse
Adjacent
Opposite
Adjacent
tan q =
cot q =
Adjacent
Opposite
sin q =
Reciprocal Relationships
cosecq =
Complementary Relationships
a = 90° - q
sin(90 - q ) = cosq cosec(90 - q ) = secq
cos(90 - q ) = sinq
sec(90 - q ) = cosecq
tan(90 - q ) = cot q
cot(90 - q ) = tanq
1
sin q
secq =
1
cosq
cot q =
1
tan q
Negative Angle
0 £ a £ 90
sin(-a ) = -sin a
cos(-a ) = cosa
tan(-a ) = - tan a
Pythagorean Identities of Trigonometry.
Using Pythagoras’ Theorem
For any angle θ
a 2 + b2 = c2
then x 2 + y2 = r 2
cos2 q + sin 2 q = 1
( r cosq )2 + ( r sinq )2 = r 2
1+ tan q = sec q
2
2
cot q +1 = cosec q
2
2
r 2 cos2 q + r 2 sin 2 q = r 2
(
)
r 2 cos2 q + sin 2 q = r 2 , divide by r 2
cos2 q + sin 2 q = 1
r
y
θ
x
Now x = r cosq and y = r sinq
cos2 q sin 2 q
1
2
+
=
, divide by sin
q
2
2
2
sin q sin q sin q
1
cot 2 q +1 = 2
sin q
cot 2 q +1 = cosec2q
Likewise,
1+ tan 2 q = sec2 q
Examples: Simplify the following
1- cos a
1- sin 2 a
b)
2
a)
=
1- (1- sin a )
1- (1- cos2 a )
2
1-1+ sin 2 a
=
1-1+ cos2 a
sin 2 a
=
cos2 a
æ sin a ö
=ç
è cos a ÷ø
= ( tan a )
= tan 2 a
2
Write down the identities
cos2 q + sin 2 q = 1
Write down the identities
Sometimes, we need to take small steps!
cos q + sin q = 1
Use the 3rd identity to replace denominator
2
Options:
(1) replace the ‘1’ with a trig expression
(2) Rearrange an identity and replace
2 cot a
1 + cot 2 a
2
1+ tan q = sec q
2
2
cot 2 q +1 = cosec2 q
In this case, rearrange the 1st identity
sin2θ = 1 – cos2θ, and cos2θ = 1 – sin2θ
2 cot a
=
cosec 2a
1+ tan 2 q = sec2 q
cot 2 q +1 = cosec2 q
Now, replace cot and cosec with
their sin and cos equivalents.
cos a
= sin a 2
æ 1 ö
çè
÷
sin a ø
2
Fraction rearrange
cosa sin 2 a
=2
´
sin a
1
1
tan q
1
secq =
cosq
1
cosecq =
sin q
cot q =
= 2 cosa sin a
Extension student would continue to the next step.
2
= sin 2a
Examples: Simplify the following
(
)
c) sec 2 a -1 tan ( 90° - a ) Write down the identities
cos2 q + sin 2 q = 1
Use the 2 identity, rearranged.
nd
= tan 2 a tan ( 90° - a )
= tan a cot a
1
= tan 2 a ´ tan a
= tan a 2
1+ tan 2 q = sec2 q
cot 2 q +1 = cosec2 q
Use the complementary trig angles.
tan(90 - q ) = cot q
Use the reciprocal trig angles.
cot q =
1
tan q
Write down the identities
d) sin3 a + sin a cos2 a
cos2 q + sin 2 q = 1
No matches, FACTORISE!
1+ tan 2 q = sec2 q
(
= sin a sin 2 a + cos2 a
Now use an identity (try number 1).
= sin a (1)
= sin a
)
cot 2 q +1 = cosec2 q
Exercise
a) Simplify
1+ cos a
sin a
sin a
1- cos a
ans : 0
b) Simplify
cosec 2a - cot 2 a
cos2 a
ans : sec2 a
c) Simplify
1
cot a + tan a
1
ans : sin a cosa = sin 2a
2
d) Simplify
1
-
1
sec a -1 sec a +1
ans : 2cot 2 a
Trigonometric Identity Proofs.
a) Prove that
cos a
- tan a = sec a
1- sin a
cos a
Break into terms of sin and cos
- tan a
1- sin a
cos a
sin a
=
Common denominator.
1- sin a cosa
cos2 a - sin a (1- sin a )
Expand numerator
=
cos a (1- sin a )
cos2 a - sin a + sin 2 a
Rearrange numerator
=
cos a (1- sin a )
cos2 a + sin 2 a - sin a
Write down the identities
=
cos a (1- sin a )
cos2 q + sin 2 q = 1
LHS =
1- sin a
=
cos a (1- sin a )
=
1
cos a
= seca
1+ tan 2 q = sec2 q
cot 2 q +1 = cosec2 q
Trigonometric Identity Proofs.
b) Prove tan a sin a + cosa = sec a
LHS = tan a sin a + cosa
sin a
=
sin a + cos a
cos a
sin 2 a + cos2 a
=
cos a
1
=
cos a
= seca
Break into terms of sin and cos
Common denominator.
Write down the identities
cos2 q + sin 2 q = 1
1+ tan 2 q = sec2 q
cot 2 q +1 = cosec2 q
Trigonometric Identity Proofs.
d) Prove sin 2 a tan a + cos2 a cot a + 2sin a cosa = tan a + cot a
LHS = sin 2 a tan a + cos2 a cot a + 2sin a cosa Break into terms of sin and cos and rearrange
sin a
cosa
= sin 2 a
+ sin a cosa + cos2 a
+ sin a cosa Factorise
cosa
sin a
æ sin 2 a
ö
æ cos2 a
ö
Express brackets as a common denominator.
= sin a ç
+ cos a ÷ + cos a ç
+ sin a ÷
è cosa
ø
è sin a
ø
æ sin 2 a + cos2 a ö
æ cos2 a + sin 2 a ö
= sin a ç
÷ø + cos a çè
÷ø
cosa
sin a
è
æ 1 ö
æ 1 ö
= sin a ç
+
cos
a
çè
÷
è cosa ÷ø
sin a ø
sin a cos a
Use definitions
=
+
cos a sin a
Use identity
Expand brackets
= tan a + cot a
This was NOT an easy question!
Exercise
2
1tan
a
a) Prove
= cos2 a - sin 2 a
2
1+ tan a
b) Prove
1+ tan 2 a
= tan 2 a
2
1+ cot a
c) Prove sin a cosa =
d) Prove
cot a
1+ cot 2 a
sin a cosa
+
= sec a coseca
cosa sin a