Answer:
tan(α+β) ≈ -5.8688
Step-by-step explanation:
You can use the formula for the tangent of the sum of angles, or you can use a calculator to find the angles. The latter is much simpler.
Calculator solution
tan(α+β) = tan(arccos(0.871) +arccos(0.338))
tan(α+β) ≈ -5.8688
Note that the two angles total slightly more than 90°, so the tangent of their sum is a large negative number.
Trig identity solution
The use of trig identities also requires a calculator, and many more steps.
The tangent of the angle can be found from the cosine using ...
tan(x) = √((1/cos(x))² -1)
tan(α) = √((1/0.871)² -1) ≈ √(1.1481056² -1) ≈ 0.56404479
tan(β) = √((1/0.338)² -1) = √(2.9585799² -1) ≈ 2.7844559
And the tangent of the sum of angles is ...
tan(α +β) = (tan(α) +tan(β))/(1 -tan(α)tan(β))
tan(α+β) = ((0.56404479 +2.7844559)/(1 -(0.56404479)(2.7844559))
tan(α+β) = 3.3485007/(1 -1.5705579)
tan(α+β) ≈ -5.8688
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Additional comment
The approximate angles are ...
α ≈ 29.4249°
β ≈ 70.2448°
Then α+β ≈ 99.6699°
