What is trigonometry?
Trigonometry is the branch of geometry that gives the relationship between the angles and sides of a triangle.
c) proof. if cos∅ = [tex]\frac{x}{\sqrt{x^{2} + y^{2} } }[/tex], then x = adjacent, [tex]\sqrt{x^{2} + y^{2} }[/tex] = hypotenuse
To solve for opposite, since it is a right angle triangle
[tex]hypotenuse^{2}[/tex] = [tex]opposite^{2}[/tex] + [tex]adjacent^{2}[/tex]
[tex](\sqrt{x^{2} + y^{2} }) ^{2}[/tex] = [tex]opposite^{2}[/tex] + [tex]x^{2}[/tex]
[tex]opposite^{2}[/tex] = [tex]x^{2}[/tex] +[tex]y^{2}[/tex] -[tex]x^{2}[/tex]
opposite = y
xsin∅ = x[tex](\frac{y}{\sqrt{x^{2} + y^{2} } })[/tex] = [tex]\frac{xy}{\sqrt{x^{2} } + y^{2} }[/tex]
ycos∅ = [tex]\frac{xy}{\sqrt{x^{2} } + y^{2} }[/tex]
Therefore xsin∅ = ycos∅
d) 41 sin∅ = 40
sin∅ = 40/41
solving for adjacent = 9
tan ∅ = opp/adj = 40/9
[tex]\frac{tan \alpha }{tan^{2}\alpha - 1 }[/tex] = 40/9 ÷ [tex](\frac{40}{9} )^{2}[/tex] - 1 = 40/9 ÷ 1600/81 - 1
= 40/9 ÷ 1519/81 = 40/9 x 81/1519 = 360/1519
e) 1 - cos ∅ = 1/2
cos ∅ = 1 - 1/2 = 1/2
∅ = cos inverse of 0.5 = 60°
tan 60 = [tex]\sqrt{3}[/tex] [tex]tan^{2} 60[/tex] = 3, sin 60 = [tex]\frac{\sqrt{3} }{2}[/tex] [tex]sin^{2} 60[/tex] = 3/4
[tex]tan^{2} 60[/tex] + 4[tex]sin^{2} 60[/tex] = 3 + 4(3/4) = 3 + 3 = 6
Learn more about trigonometry: brainly.com/question/24349828
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