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chapter4.6r
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à 4.6ïProduct and Sum Identities
äïPlease use the Product or Sum Identities to answer the
êêfollowing questions.
âè Evaluate sin 5π/12 + sin π/12 by using a Sum Identity.
êësin A + sin Bï=ï2∙sin (A + B)/2 ∙ cos (A - B)/2
sin 5π/12 + sin π/12ï=ï2∙sin (5π/12 + π/12)/2 ∙ cos (5π/12 - π/12)/2
êï=ï2∙sin π/4 ∙ cos π/6ï=ï2∙ 1/√2 ∙ √3/2ï=ï√3/√2
éSïThere are some additional identities that are useful on a few
occasions in later math courses.ïThey are the product-to-sum identities
and the sum-to-product identities.ïThese formulas can be derived by
adding or subtracting the Addition Identities.ïFor example, you can
add the formulas for cos (A + B) and cos (A - B).
cos (A + B) + cos (A - B)ï=ïcos A∙cos B - sin A∙sin B + cos A∙cos B +
sin A∙sin Bï=ï2∙cos A∙cos B.ïThus, solving for cos A∙cos B,
êë cos A∙cos Bï=ï1/2∙[cos (A + B) + cos (A - B)].
The complete list of sum-to-product and product-to-sum identities are
listed below.
êêêïSum-to-Product Identities
êè1)ïsin A + sin Bï=ï2∙sin (A + B)/2 ∙ cos (A - B)/2
êè2)ïsin A - sin Bï=ï2∙cos (A + B)/2 ∙ sin (A - B)/2
êè3)ïcos A + cos Bï=ï2∙cos (A + B)/2 ∙ cos (A - B)/2
êè4)ïcos A - cos Bï=ï-2∙sin (A + B)/2 ∙ sin (A - B)/2
êêêïProduct-to-Sum Identities
êè5)ïsin A∙cos Aï=ï1/2[sin (A + B) + sin (A - B)]
êè6)ïcos A∙cos Bï=ï1/2[sin (A + B) - sin (A - B)]
êè7)ïsin A∙sin Bï=ï1/2[cos (A - B) - cos (A + B)]
êè8)ïcos A∙cos Bï=ï1/2[cos (A + B) + cos (A - B)]
Also included here are the Cofunction Identities and the Opposite-Angle
Identities.
èCofunction IdentitiesêêïOpposite-Angle Identities
9)ïcos (π/2 - x)ï=ïsin xêë 12)ïcos (-x)ï=ïcos x
10)ïsin (π/2 - x)ï=ïcos xêë13)ïsin (-x)ï=ï-sin x
11)ïtan (π/2 - x)ï=ïcot xêë14)ïtan (-x)ï=ï-tan x
1êUse a sum-to-product identity to evaluate
êêêë sin 105° - sin 15°
êê A)ï2êêêëB)ï1/√2
êê C)ï√3/2êêêïD)ïå of ç
ü
êë sin 105° - sin 15°ï=ï2∙cos 120°/2 ∙ sin 90°/2
êêêêë= 2∙cos 60° ∙ sin 45°
êêêêë=ï2∙ 1/2 ∙1/√2
êêêêë=ï1/√2
Ç B
2êUse a sum-to-product identity to evaluate
êêêë sin 165° + sin 75°
êê A)ï1/2êêêèB)ï√3
êê C)ï√3/√2êêê D)ïå of ç
ü
êë sin 165° + sin 75°ï=ï2∙sin 240°/2 ∙ cos 90°/2
êêêêë= 2∙sin 120° ∙ cos 45°
êêêêë=ï2∙ √3/2 ∙1/√2
êêêêë=ï√3/√2
Ç C
3êUse a sum-to-product identity to evaluate
êêêë cos 5π/12 + cos π/12
êê A)ï√3/√2êêê B)ï√3
êê C)ï1/2êêêèD)ïå of ç
ü
êë cos 5π/12 + cos π/12ï=ï2∙cos 6π/12/2 ∙ cos 4π/12/2
êêêêë= 2∙cos π/4 ∙ cos π/6
êêêêë=ï2∙ 1/√2 ∙√3/2
êêêêë=ï√3/√2
Ç A
4êUse a sum-to-product identity to evaluate
êêêë cos 42° - cos 28°
êê A)ï.2781êêê B)ï√3
êê C)ï.1398êêê D)ïå of ç
ü
êë cos 42° - cos 28°ï=ï-2∙sin 70°/2 ∙ sin 14°/2
êêêêë= 2∙sin 35° ∙ sin 7°
êêêêë≈ï2∙(.5736)∙(.1219)
êêêêë≈ï.1398
Ç C
5êUse a product-to-sum identity to evaluate
êêêë cos 3π/8 ∙ cos π/8
êê A)ï1/√2êêêïB)ï√2/4
êê C)ï1/√3êêêïD)ïå of ç
ü
êë cos 3π/8 ∙ cos π/8ï=ï1/2[cos 4π/8 + cos 2π/8]
êêêêë= 1/2[cos π/2 + cos π/4]
êêêêë=ï1/2[ 0 + 1/√2]
êêêêë=ï1/2√2ï=ï√2/4
Ç B
6êUse a product-to-sum identity to evaluate
êêêë sin 55° ∙ cos 35°
êê A)ï.2897êêê B)ï.1823
êê C)ï.6710êêê D)ïå of ç
ü
êë sin 55° ∙ cos 35°ï=ï1/2[sin 90° + sin 20°]
êêêêë≈ 1/2[ 1 + .3420]
êêêêë≈ï.6710
Ç C
7ê Prove one of the following is an identity.
êë A)ï2∙cos 3x/2 ∙ sin x/2ï=ïsin 2x - sin x
êë B)ïcsc x - tan x/2ï=ïsec x
êë C)ïå of ç
ü
êêêè2∙cos 3x/2 ∙ sin x/2
êêë=è 2∙cos (2x + x)/2 ∙ sin (2x - x)/2
êêë=è sin 2x - sin x
Ç A
8ê Prove one of the following is an identity.
#êë A)ï(sin 5x + sin 3x)/(cos 5x + cos 3x)ï=ïsinì4x
êë B)ï(cos 3x - cos x)/(sin x - sin 3x)ï=ïtan 2x
êë C)ïå of ç
üêêè cos 3x - cos x
#êêêè──────────────
êêêèsin x - sin 3x
êêêè-2∙sin 4x/2 ∙ sin 2x/2
#êêë=è ──────────────────────
êêêè2∙cos 4x/2 ∙ sin 2x/2
êêêèsin 2x
#êêë=è ──────
êêêècos 2x
êêë=è tan 2x
Ç B