3.5 DOUBLE-ANGLE AND HALF-ANGLE FORMULAS
DOUBLE ANGLE FORMULAS
By substituting α = β = θ into the sin(α+β),cos(α+β) and tan(α+β) formulas
we get:
sin(2θ) = 2sinθcosθ
cos(2θ) = cos2θ – sin2θ
If we use the identity that cos2θ + sin2θ =1, then we can substitute
cos2θ = 1 - sin2θ into the cos(2θ) formula for another variation:
cos(2θ) = (1 - sin2θ) – sin2θ = 1 – 2sin2θ
Substituting sin2θ = 1 - cos2θ gives the following:
cos(2θ) = cos2θ – (1 - cos2θ) = 2 cos2θ - 1
tan(2θ) =
2 tan θ
1 − tan 2 θ
Example 1
If sin θ = 3/5, and π/2 < θ < π, find the exact value of:
θ is in Quadrant II so cos θ is < 0.
3
5
52 − 32 = 16 = 4
a) sin(2θ)
We know sin θ, and from the right triangle and the fact that θ is in
Quadrant II, cos θ = -4/5
sin(2θ) = 2sin θ cos θ = 2(3/5)(-4/5) = -24/25
b) cos(2θ) = 1 – 2sin2θ
= 1-2(3/5)2 = 1 – 2(9/25) = 1 – 18/25 = 7/25
Now do # 1 a and b
Example 4
An object is propelled upward at an angle θ to the horizontal
with an initial velocity of v0 feet per second. If air resistance
is ignored, the range, R, this horizontal distance that the
object travels, is given by
1 2
R = v0 sin θ cos θ
16
θ
R
A) Show that
1 2
R = v0 sin (2θ )
32
B) Find the angle θ for which R is a maximum
a) Use double angle formula -> sin(2θ) = 2 sin θ cos θ
Divide both sides by 2
½ sin(2θ) = sin θ cos θ
Substitute this into the Range formula
1 2
1 21
R = v0 sin( 2θ ) = v0 sin( 2θ )
32
16 2
b) The maximum Range will occur at the max sin value which is
1. For what angle θ will sin(2θ) = 1?
sin-1(sin2θ) = sin-1 1
2θ = π/2 rad = 90°
1 2
v0 (1)
θ = 45° will give a maximum range of R =
32
Variations of the Double Angle Formula
1 − cos(2θ )
2
sin θ =
2
1 + cos(2θ )
2
cos θ =
2
1 − cos(2θ )
2
tan θ =
1 + cos(2θ )
HALF-ANGLE FORMULAS
α
1 − cos α
sin = ±
2
2
α
1 + cos α
cos = ±
2
2
α
1 − cos α
tan = ±
2
1 + cos α
We need to know
which Quadrant
α/2 is in order to
choose the + or –
answer.
We can use these formulas to find the exact value of trig functions
of angles that when doubled become a more familiar angle from
our Unit Circle.
Examples 5 and 6 will be done in class.
HOMEWORK for 3.5
p.241 #1 ,5, 11,13, 15, 23, 51, 53, 63, 69, 77