Six trig functions in one diagram
The diagram shows a circle, centre O, radius 1 with tangent, BC, at A.
l = BOC
l = 90D
ADO
C
A
1
θ
O
D
B
1. Explain why the angle at C is also θ .
2. Match up the six trigonometric functions:
sin θ ,
cos θ ,
tan θ ,
cot θ =
1
,
tan θ
secθ =
1
,
cos θ
cosecθ =
1
sin θ
with the lengths OB, OC, OD, AB, AC, AD. Give reasons for each one.
3. Three lengths have not been considered so far: BC, BD and CD.
(a) Consider the length BC:
BC = BA +AC
and BC2 = OB2 + OC2
What trigonometric identities are lurking here?
(b) Investigate the length BD.
© MEI 2006
What does the graph y = sin x + cos x look like?
What does the graph y = sin x × cos x look like?
Four handy diagrams for deriving compound or double angles
Circle theorems
Rotated rectangle
Chord in unit circle
Area of triangle
Compound Angle Formulae
The textbook explains how the six compound angle formulae can be derived:
sin ( A ± B ) = sin A cos B ± cos A sin B
cos ( A ± B ) = cos A cos B ∓ sin A sin B
tan ( A ± B ) =
tan A ± tan B
1 ∓ tan A tan B
Substituting A = B in the above formulae leads to the double angle formulae:
sin 2 A = 2sin A cos A
cos 2 A = cos 2 A − sin 2 A
2 tan A
tan 2 A =
1 − tan 2 A
These can all be proved using diagrams, assuming you’re prepared to think
carefully about them! A few are on the following pages with some reasoning
left for you to supply.
© MEI 2007
Proving the double angle formulae
1
2θ
θ
1
ˆ = θ then, by the circle theorems, DOC
ˆ = 2θ and ACB
ˆ = 90 .
If OAC
From the right-angled triangle ABC , cos θ =
AC
AB
⇒
AC = 2 cos θ
Therefore, From the right-angled triangle ACD ,
CD
sin θ =
⇒ CD = 2 cos θ sin θ
AC
But from the right-angled triangle OCD , sin 2θ =
CD
OC
⇒ CD = sin 2θ .
Combining these we have CD = sin 2θ = 2cos θ sin θ
•
Can you use the same diagram to prove the three identities for cos 2θ ?
cos 2θ ≡ cos 2 θ − sin 2 θ ≡ 2 cos 2 θ − 1 ≡ 1 − 2 sin 2 θ
© MEI 2007
How does this tilted rectangle prove the formulae given?
1
See the Geogebra file Compound angle 2
Use this diagram to prove that tan (α + β ) =
BE tan α + tan β
=
AE 1 − tan α tan β
© MEI 2007
Proving cos ( x − y ) = cos x cos y + sin x sin y using rotations and the unit circle.
y
1
Take two points P
and Q fixed on the
unit circle as shown
P : ( cos x,sin x )
x
Q : ( cos y,sin y )
x
y
–1
–1
1
–1
The length of chord PQ is given by:
PQ = ( cos y − cos x ) + ( sin x − sin y )
2
2
2
= cos 2 y − 2 cos x cos y + cos 2 x + sin 2 x − 2sin x sin y + sin 2 y
= 2 − 2 ( cos x cos y + sin x sin y )
Now rotate the circle together with the points through an angle y in a
clockwise direction. The result is shown below:
y
1
P : ( cos ( x − y ) ,sin ( x − y ) )
Q : (1, 0x)
x-y
–1
–1
1
–1
The chord PQ still has the same length but now the length is given by:
PQ = (1 − cos ( x − y ) ) + ( sin ( x − y ) )
2
2
2
= 1 − 2 cos ( x − y ) + cos 2 ( x − y ) + sin 2 ( x − y )
= 2 − 2 cos ( x − y )
2
Equating the two expressions for PQ completes the proof.
© MEI 2007