Answer:
Thus, option (c) is correct.
β = 61.5
Step-by-step explanation:
Given , [tex]\sin (\frac{x}{2}+20x)=\cos(2x+\frac{15x}{2})[/tex] , we have to solve for x, and then find the value of β ( β > α )
Consider [tex]\sin (\frac{x}{2}+20x)=\cos(2x+\frac{15x}{2})[/tex],
First solve for x ,
[tex]\Rightarrow \sin (\frac{x}{2}+20x)=\cos(2x+\frac{15x}{2})[/tex]
[tex]\Rightarrow \sin (\frac{x+40x}{2})=\cos(\frac{4x+15x}{2})[/tex]
Thus, [tex]\Rightarrow \sin (\frac{41x}{2})=\cos(\frac{19x}{2})[/tex]
Also, [tex]\sin (90-\theta)=\cos \theta[/tex] , we get,
Thus, [tex]\Rightarrow \cos (90-\frac{41x}{2})=\cos(\frac{19x}{2})[/tex]
since, LHS = RHS thus, angle must be equal,
[tex]\Rightarrow 90-\frac{41x}{2}=\frac{19x}{2}[/tex]
[tex]\Rightarrow 90=\frac{19x}{2}+\frac{41x}{2}[/tex]
[tex]\Rightarrow 90=\frac{19x+41x}{2}[/tex]
[tex]\Rightarrow 90=\frac{60x}{2}[/tex]
[tex]\Rightarrow x=3[/tex]
Thus, [tex]\frac{x}{2}+20x=\frac{3}{2}+20(3)=\frac{3}{2}+60=61.5[/tex] ,
Other angle can be found using angle sum property, as sum of angle of a triangle is 180°
Let third angle be y, then ,
90 + 61.5 + y = 180°
y = 180° - 151.5°
y = 28.5°
Since ( β > α ) ⇒ β= 61.5 and α = 28.5
Thus, option (c) is correct.
⇒ β = 61.5
