find exact value of cos(tan^-1 4/3 + cos^-1 5/13)

Answer 1

First, let's make an assumption that we are working in Q1 (we'll revisit that assumption later).

Let's let α be the angle for which the tan is 4/3 and let's let β be the angle for which cos is 5/13.

Considering angle α, α = tan^-1(4/3) and the associated triangle would look like this:

because tan(α) = opposite/adjacent. This allows us to solve for "x" using the Pythagorean Theorem. x = 5

Considering angle β, β = cos^-1(5/13) and the associated triangle is:

because cos(β) = adjacent/hypotenuse. This allows us the solve for y using the Pythagorean Theorem. y = 12

The cosine angle addition identity says:

cos(α + β) = cos(α)cos(β) - sin(α)sin(β)

Using the triangle we got for angle α we can say cos(α) = 3/5 and sin(α) = 4/5

Using the triangle we got for angle β we can say cos(β) = 5/13 and sin(β) = 12/13

So cos(α + β) = (3/5)(5/13) - (4/5)(12/13) = 15/65 - 48/65 = -33/65

Going back to our initial assumption of being in Q1, angle α could also be in Q3 and be positive and angle β could also be in Q4 and be positive. This would mean cos(α) could be -3/5 and sin(α) could be -4/5). It also means cos(β) could only be 5/13 but that sin(β) could be -12/13. 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 variety of combinations of the above values, you can get both ± 33/65 and ± 63/65 for your answers so it really depends on which quadrant you are in.

However, using the "function" definitions of tan^-1 (where the angle is in either Q1 or Q4) and cos^-1 (where the angle is in either Q1 or Q2) it is reasonable to use the Q1 definition that we started with to get -33/65.