Download 11.5A Half-Angle Identities

11.5A Half-Angle Identities
Objective:
F.TF.9: Prove the addition and subtraction formulas for sine, cosine, tangent and use them to
solve problems.
For the Board: You will be able to evaluate and simplify expressions by using double-angle and halfangle identities.
Anticipatory Set:
Let θ = α/2, then
cos 2θ = 1 – 2 sin2 θ becomes cos 2(α/2) = 1 – 2 sin2 (α/2)
cos α = 1 – 2 sin2 (α/2)
2 sin2 (α/2) = 1 – cos α
sin2 (α/2) = (1 – cos α)/2
1  cos α
sin α/2 = 
2
Instruction:
Half-Angle Identities
sin α/2 = 
1  cos α
1  cos α
1  cos α
cos α/2 = 
tan α/2 = 
2
2
1  cos α
Choose + or – depending on the location of α/2.
Open the book to page 786 and read example 3.
Example: Use half-angle identities to find the exact value of each trigonometric expression.
a. cos 15°
b. tan 7π/8
15° is half of 30°
cos 15° =
15° is in quadrant I so cos 15° is positive
1  cos α
1  cos 30
=
=
2
2
7π/8 is half of 14π/8 or 7π/4
tan 7π/8 = 
=
1  3/2

2
2 3

4
2 3
2
7π/4 is in quadrant IV so tan 7π/4 is negative
1  cos α
1  2 /2
2 2 2 2
44 2 2

=
=
=
42
1  cos α
1  2 /2
2 2 2 2
44 2 2
64 2
=
= 3 2 2
42
2
White Board Activity:
Practice: Use half-angle identities to find the exact value of each trigonometric expression.
a. tan 75°
b. cos 5π/8
75° is half of 150°
tan 75° =
=
75° is in quadrant I so tan 75° is positive.
1  cosα
1  cos150
1  3 /2
2 3 2 3
=
=
=

1  cos α
1  cos150
1  3 /2
2 3 2 3
44 33
= 74 3
43
5π/8 is half of 5π/4
cos 5π/8 = 
=
5π/4 is in quadrant III so cos 5π/8 is negative.
1  cos α
1  cos 5/4
1  2 /2
2 2
=
=
=
2
4
2
2
2 2
2
Open the book to page 786 and read example 4.
Example: Find cos θ/2 and tan θ/2 if tan θ = 7/24 and 0 < θ < π/2.
7
θ
24
72 + 242 = r2 49 + 576 = r2 r2 = 625 r = 25
Since cos θ is in quadrant I so is cos θ/2.
cos θ = 24/25
1  cosθ
1  24/25
49
7 2
7
cos θ/2 =
=
=
=
=
2
2
50 5 2
10
tan θ/2 =
1  cos α
1  24/25
1/25
1
=
=
=
= 1/7
1  cos α
1  24/25
49/25
49
White Board Activity:
Practice: Find sin θ/2 and cos θ/2 if tan θ = 4/3 and 0° < θ < 90°.
32 + 42 = r2 9 + 16 = r2 r2 = 25 r = 5
Since cos θ is in quadrant I so is cos θ/2.
cos θ = 3/5
4
1  cosθ
1  3/5
2
1
5
sin θ/2 =
=
=
=
=
θ 3
2
2
10
5
5
1  cosθ
1  3/5
8
4 2 5
cos θ/2 =
=
=
=
=
2
2
10
5
5
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 788 prob. 7 – 12, 19 - 24.