Answer:
[tex]\boxed{\boxed{\beta=15^{\circ}}}[/tex]
Step-by-step explanation:
As given as ∠α and ∠β are the two acute angles in a right triangle.
So,
[tex]\Rightarrow \alpha+\beta=90^{\circ}[/tex]
[tex]\Rightarrow \alpha=90^{\circ}-\beta[/tex]
[tex]\Rightarrow \sin \alpha=\sin(90^{\circ}-\beta)[/tex]
[tex]\Rightarrow \sin \alpha=\cos \beta[/tex]
Also given as,
[tex]\sin(3x - 27) = \cos(5x + 5)[/tex]
Then between (3x-27) and (5x+5), one is α and the other one is β.
And the sum of both the angles are 90°. So,
[tex]\Rightarrow (3x - 27)+(5x + 5)=90[/tex]
[tex]\Rightarrow 8x - 22=90[/tex]
[tex]\Rightarrow 8x=90+22=112[/tex]
[tex]\Rightarrow x=14[/tex]
Then the measurement of the two angles are,
[tex]3x - 27=3(14) - 27=15^{\circ}\\\\5x + 5=5(14) + 5=75^{\circ}[/tex]
As β < α, so β=15°
