can someone please show me how to do this math problem tonight?

Question

evaluate:
cos[Arctan (-2)]

Answer 1

One way of doing it is just looking up arctan in a table and then applying cosine to the result.

Another way is to use the trigonometric functions directly

tan(x) = sin(x)/cos(x)

arctan(y) is the angle whose tangent is y

so cos(arctan(y)) is the cosine of the angle whose tangent is y

but we also know that: (cos(x))^2 + (sin(x))^2 = 1

so except for sign:

sin(x) = sqrt(1 - (cos(x))^2)

so, up to a sign change, letting u = cos(x):

tan(x) = sqrt(1 - (cos(x))^2)/cos(x)
tan(x) = sqrt(1 - u^2)/u or
u tan(x) = sqrt(1 - u^2)

we know the value of tan(x) - it is just -2 -so we can square both sides to get a simple quadratic. Then we can solve the quadratic and adjust the sign.