First, let"s do an assumption that we space working in Q1 (we"ll revisit that assumption later).
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Let"s permit α it is in the angle for which the tan is 4/3 and let"s permit β it is in the angle because that which cos is 5/13.
Considering edge α, α = tan-1(4/3) and also the associated triangle would certainly look like this:
because tan(α) = opposite/adjacent. This enables us to solve for "x" using the Pythagorean Theorem. X = 5
Considering edge β, β = cos-1(5/13) and the linked triangle is:
because cos(β) = adjacent/hypotenuse. This enables us the solve for y making use of the Pythagorean Theorem. Y = 12
The cosine angle enhancement identity says:
cos(α + β) = cos(α)cos(β) - sin(α)sin(β)
Using the triangle we obtained for edge α we deserve to say cos(α) = 3/5 and also sin(α) = 4/5
Using the triangle we obtained for edge β we can say cos(β) = 5/13 and also sin(β) = 12/13
So cos(α + β) = (3/5)(5/13) - (4/5)(12/13) = 15/65 - 48/65 = -33/65
Going ago to our initial presumption of being in Q1, angle α could additionally be in Q3 and also be positive and also angle β could also be in Q4 and also be positive. This would mean cos(α) could be -3/5 and sin(α) might be -4/5). It also method cos(β) could only it is in 5/13 however that sin(β) can be -12/13.
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Putting these values into the equation gives:
cos(α + β) = (-3/5)(5/13) - (-4/5)(-12/13) = -15/65 - 48/65 = -63/65
Using a selection of combinations of the above values, you can acquire both ± 33/65 and also ± 63/65 for your answers so the really counts on i beg your pardon quadrant you room in.
However, utilizing the "function" meanings of tan-1 (where the angle is in either Q1 or Q4) and also cos-1 (where the edge is in either Q1 or Q2) it is reasonable to use the Q1 definition that we started with to get -33/65.