Answer:
[tex]\tan^{-1}\left(\dfrac{40}{100}\right)-\tan^{-1}\left(\dfrac{20}{100}\right)[/tex]
Step-by-step explanation:
You want the difference between the angle to a point 40 ft high and the angle to a point 20 ft high when both are 100 horizontal feet from your location.
Tangent
The tangent ratio is ...
Tan = Opposite/Adjacent
This tells you the tangent of the angle to the top of the stage is ...
tan(angle to top) = (40 ft)/(100 ft)
Similarly, the tangent of the angle to the bottom of the stage is ...
tan(angle to bottom) = (20 ft)/(100 ft)
Angles
The angles are found using the inverse tangent function:
angle to top = arctan(40/100)
angle to bottom = arctan(20/100)
The angle theta is the difference of these:
θ = (angle to top) -(angle to bottom)
θ = arctan(40/100) -arctan(20/100)
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Additional comment
The fractions can be reduced. You want an exact expression, and that is what this is. Since the calculator will be used to find the angle, we can let it reduce the fraction as it may need. (Be sure the calculator is using the angle mode you want. Usually, that will be degrees for a problem like this.)
The inverse tangent function is also called the "arctangent" function. On most calculator keyboards, the function has a -1 exponent, signifying the inverse function. The attachment shows θ ≈ 10.5°.
