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] + 4x
Or, 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] α
So, As 37.5°[tex]<[/tex]52.5°
∴ β = 37.5°
Hence From given relation the value of β is 37.5° Answer
