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BGSU
5.3 Double-Angle, Power-Reducing, and Half-Angle Formulas
Math 1300
Double-Angle Formulas
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ
• tan 2θ =
2 tan θ
1−tan2 θ
Example 1 Use the given information to find the exact value of each of the following:
a. sin 2θ
b. cos 2θ
c. tan 2θ
1. (#8) sin θ =
12
13 , θ
lies in quadrant II.
2. (#10) cos θ =
40
41 , θ
Example 2 Find the exact value of the expression.
1. (#16) 2 sin 22.5◦ cos 22.5◦
2. (#20) 1 − 2 sin2
Example 3 (#28) Verify the identity: 1 − tan2 x =
lies in quadrant IV.
π
12
cos 2x
.
cos2 x
Power-Reducing Formulas
sin2 θ =
1−cos 2θ
2
Ying-Ju Tessa Chen
cos2 θ =
1+cos 2θ
2
Last modified: November 8, 2014
tan2 θ =
1−cos 2θ
1+cos 2θ
1
BGSU
5.3 Double-Angle, Power-Reducing, and Half-Angle Formulas
Math 1300
Example 4 (P. 648 Check Point 4) Write an equivalent expression for sin4 x that does not contain powers of trigonometric functions greater than 1.
Example 5 Verify each identity.
1. (P. 650 Check Point 6) tan θ =
sin 2θ
1+cos 2θ
2. (#30) cot x =
1+cos 2x
sin 2x
Half-Angle Formulas
q
α
sin α2 = ± 1−cos
2
q
α
tan α2 = ± 1−cos
1+cos α =
q
α
cos α2 = ± 1+cos
2
1−cos α
sin α
=
sin α
1+cos α
Remark 1 The ± symbol in each formula does not mean that there are two possible values for each
function. We determine the sign of the trigonometric function, + or −, based on the quadrant in which
the half-angle α2 lies.
Example 6 Find the exact value of
1. (#40) cos 22.5◦
2. (#42) sin 105◦
Example 7 (#62) Verify each identity cos2
θ
2
=
3. (#46) tan 3π
8
sec θ+1
2 sec θ
Exercise 1 P. 652 # 36, 38, 44, 48, 50, 54, 56, 60, 64, 66
Ying-Ju Tessa Chen
Last modified: November 8, 2014
2