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Find the exact value of each of the following under the given conditions. sine alpha equals eight seventeenths comma 0 less than alpha less than StartFraction pi Over 2 EndFraction ​; cosine beta equals StartFraction 6 StartRoot 61 EndRoot Over 61 EndFraction comma negative StartFraction pi Over 2 EndFraction less than beta less than 0 ​(a) sine (alpha plus beta )​(b) cosine (alpha plus beta )​(c) sine (alpha minus beta )​(d) tangent (alpha minus beta )

Respuesta :

Answer:

a) Sin (α + β) = [123/(17√61)] = 123 ÷ (17√61)

b) Cos (α + β) = [50/(17√61)] = 50 ÷ (17√61)

c) Sin (α - β) = [-27/(17√61)] = -27 ÷ (17√61)

d) Tan (α - β) = (-27/130)

Step-by-step explanation:

Sin α = (8/17) for (0 < α < π/2)

Cos β = [(6√61)/61] = (6/√61) for (0 < β < π/2)

Note that

Sin²x + Cos²x = 1

Sin²α + Cos²α = 1

(8/17)² + Cos²α = 1

Cos²α = 1 - (64/289) = (225/289)

Cos α = (15/17)

Sin²β + Cos²β = 1

Sin²β + (6/√61)² = 1

Sin²β = 1 - (36/61) = (25/61)

Sin β = (5/√61)

Sin α = (8/17)

Cos α = (15/17)

Sin β = (5/√61)

Cos β = (6/√61)

a) Sin (α + β) = Sin α Cos β + Sin β Cos α

= (8/17) × (6/√61) + (5/√61) × (15/17)

= [48/(17√61)] + [(75/(17√61)]

= [123/(17√61)]

= 123 ÷ (17√61)

b) Cos (α + β) = Cos α Cos β - Sin α Sin β

= (15/17) × (6/√61) - (8/17) × (5/√61)

= [90/(17√61)] - [40/(17√61)]

= [50/(17√61)]

= 50 ÷ (17√61)

c) Sin (α - β) = Sin α Cos β - Sin β Cos α

= (8/17) × (6/√61) - (5/√61) × (15/17)

= [48/(17√61)] - [(75/(17√61)]

= [-27/(17√61)]

= -27 ÷ (17√61)

d) Tan (α - β) = [Sin (α - β)] ÷ [Cos (α - β)]

Sin (α - β) = [-27/(17√61)] = -27 ÷ (17√61)

Cos (α - β) = Cos α Cos β + Sin α Sin β

= (15/17) × (6/√61) + (8/17) × (5/√61)

= [90/(17√61)] + [40/(17√61)]

= [130/(17√61)]

= 130 ÷ (17√61)

Tan (α - β) = [Sin (α - β)] ÷ [Cos (α - β)]

= [-27/(17√61)] ÷ [130/(17√61)]

= (-27/130)

Hope this Helps!!!

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