Answer:
A) 141.3°
Step-by-step explanation:
To solve the given trigonometric equation for β, begin by rearranging the equation to isolate cos β:
[tex]\dfrac{\cos \beta}{3}+0.42=0.16\\\\\\\dfrac{\cos \beta}{3}+0.42-0.42=0.16-0.42\\\\\\\dfrac{\cos \beta}{3}=-0.26\\\\\\\dfrac{\cos \beta}{3}\times 3=-0.26\times 3\\\\\\\cos \beta=-0.78[/tex]
Since the cosine of an angle is negative in both the second and third quadrants, we will find two angles that satisfy cos(β) = -0.78.
To find the principal value of β, use the inverse cosine function on a calculator:
[tex]\beta=\cos^{-1}(-0.78)\\\\\beta=141.3^{\circ}[/tex]
As this angle is between 90° and 180°, it lies in quadrant II.
To find the other value of β in quadrant III, we subtract the principal angle from 360°:
[tex]\beta=360^{\circ}-141.3^{\circ}\\\\\beta=218.7^{\circ}[/tex]
As cosine is a periodic function, it repeats its values every 360°. Therefore, we can add 360°n (where n is an integer) to both solutions to account for the periodic nature of trigonometric functions.
So, the solutions to the given equation are:
[tex]\beta = 141.3^{\circ}+360^{\circ}n,\;\;\beta = 218.7^{\circ}+360^{\circ}n[/tex]
If we subtract 360° from both solutions, we get negative degree values. Therefore, the smallest positive degree value of β is:
[tex]\Large\boxed{\boxed{141.3^{\circ}}}[/tex]