Answer:
D
Step-by-step explanation:
Using the addition formula for cosine
cos(α + β) = cosαcosβ - sinαsinβ
Given α in quadrant 3 then and sinα = - [tex]\frac{4}{5}[/tex] = [tex]\frac{opposite}{hypotenuse}[/tex] , then
cosα < 0 in the third quadrant
The adjacent side of the triangle = 3 since 3- 4- 5 right triangle
Thus
cosα = [tex]\frac{adjacent}{hypotenuse}[/tex] = - [tex]\frac{3}{5}[/tex]
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Given β in second quadrant and sinβ = [tex]\frac{1}{2}[/tex] = [tex]\frac{opposite}{hypotenuse}[/tex] , then
cosβ < 0 in the second quadrant
The adjacent side of the triangle = [tex]\sqrt{3}[/tex] since 1- [tex]\sqrt{3}[/tex] - 2 right triangle
Thus
cosβ = [tex]\frac{adjacent}{hypotenuse}[/tex] = - [tex]\frac{\sqrt{3} }{2}[/tex]
Hence
cos(α + β )
= ( - [tex]\frac{3}{5}[/tex] × - [tex]\frac{\sqrt{3} }{2}[/tex] ) - (- [tex]\frac{4}{5}[/tex] × [tex]\frac{1}{2}[/tex] )
= [tex]\frac{3\sqrt{3} }{10}[/tex] + [tex]\frac{4}{10}[/tex]
= [tex]\frac{3\sqrt{3}+4 }{10}[/tex] → D
