Angles α and β are the two acute angles in a right triangle. Use the relationship between sine and cosine to find the value of β if β < α. sin( x/2 + 2x) = cos(2x + 3x/2 )A) 15° B) 37.5° C) 52.5° D) 75°

Answer:From given relation the value of β is 37.5°Step-by-step explanation:Given as : α and β are two acute angles of right triangle Acute angle have measure less than 90° Now given as :[tex]sin(\frac{x}{2} + 2x)[/tex] = [tex]cos(2x +\frac{3x}{2})[/tex]Or, [tex]cos(90° - (\frac{x}{2}+2x))[/tex] =  [tex]cos(2x +\frac{3x}{2})[/tex]SO, [tex](90° - (\frac{x}{2}+2x))[/tex] = [tex]2x+\frac{3x}{2}[/tex]Or, 90° =  [tex]2x+\frac{3x}{2}[/tex] + [tex]\frac{x}{2}+2x[/tex]or, 90° = [tex]\frac{4x}{2}[/tex] + 4xOr,  90° =  [tex]\frac{12x}{2}[/tex]So, x =  [tex]\frac{90}{6}[/tex] = 15°∴ [tex]sin(\frac{x}{2} + 2x)[/tex] = [tex]sin(\frac{15}{2} + 30)[/tex]So, [tex]sin(\frac{x}{2} + 2x)[/tex] = sin[tex]\frac{75}{2}[/tex] ∴  The value of Ф_1 = [tex]\frac{75}{2}[/tex] = 37.5° Similarly  [tex]cos(2x +\frac{3x}{2})[/tex] =  [tex]cos(30 +\frac{45}{2})[/tex]So ,The value of Ф_2 = [tex]\frac{105}{2}[/tex] = 52.5°∵ β  [tex]<[/tex] αSo, As 37.5°[tex]<[/tex]52.5°∴ β = 37.5° Hence From given relation the value of β is 37.5°  Answer