ACCEL. PRE-CALCULUS/TRIG 3
Double Angle Formulas:
𝑠𝑖𝑛2𝑥 = 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
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𝑐𝑜𝑠2𝑥 = 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥
= 2𝑐𝑜𝑠 2 𝑥 − 1
= 1 − 2𝑠𝑖𝑛2 𝑥
2𝑡𝑎𝑛𝑥
𝑡𝑎𝑛2𝑥 = 1−𝑡𝑎𝑛2 𝑥
Double Angle Examples:
3
1) If sin x= 4 in Quadrant I, find each of the double angle values.
a) sin 2x=
2) If cos x=
−24
25
a) sin 2x=
b) cos 2x=
c) tan 2x=
in Quadrant III, find each of the double angle values.
b) cos 2x=
c) tan 2x=
Verifying Identities Double Angle Formulas
EXAMPLES:
1) cos 4 x − sin4 x = cos2x
Deriving a Triple Angle Formula
1) sin 3x =
2) sin2x = −2sinxsin(x − 90°)
Half Angle Formulas:
𝑥
1−𝑐𝑜𝑠𝑥
𝑠𝑖𝑛 2 = ±√
2
𝑥
1+𝑐𝑜𝑠𝑥
𝑐𝑜𝑠 2 = ±√
𝑥
1−𝑐𝑜𝑠𝑥
𝑡𝑎𝑛 2 = ±√1+𝑐𝑜𝑠𝑥 =
2
Half Angle Examples:
3
1) If sin x= 5 in Quadrant II, find each of the half angle values.
𝒙
𝒙
a) sin 𝟐 =
𝒙
b) cos 𝟐=
c) tan 𝟐=
1
2) If cos x= – 2 in Quadrant III, find each of the half angle values.
𝒙
a) sin =
𝟐
Putting it all together:
1) cos 15°
𝒙
𝒙
b) cos =
c) tan =
𝟐
𝟐
2) sin 120°
1−𝑐𝑜𝑠𝑥
𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥
= 1+𝑐𝑜𝑠𝑥