What is the other trigonometric functions if the tan=1/2?

1 Answer
May 22, 2018

#cos arctan(1/2) = pm 2/sqrt{5}#

#sin arctan(1/2) = pm 1/sqrt{5}#

#cot arctan(1/2) = 2 #

#csc arctan(1/2) = pm sqrt{5} #

#sec arctan(1/2) = pm \sqrt{5}/2#

Explanation:

A tangent or slope of #1/2# is a right triangle with opposite #1#, adjacent #2# so a hypotenuse of #sqrt{1^2+2^2}=sqrt{5}.# The other trig functions are made of these components. We can't be sure of the sign of any of them except the cotangent.

#cos arctan(1/2) = pm 2/sqrt{5}#

#sin arctan(1/2) = pm 1/sqrt{5}#

# tan arctan(1/2) = 1/2 #

#cot arctan(1/2) = 1/{tan arctan(1/2)} = 2 #

#csc arctan(1/2) = pm sqrt{5} #

#sec arctan(1/2) = pm \sqrt{5}/2#

Let's work out six trig functions in general given #tan x = a/b.# That's a right triangle, opposite #a#, adjacent #b#, hypotenuse #sqrt{a^2+b^2}.#

#cos arctan(a/b) = pm b/sqrt{a^2+b^2}#

#sin arctan(a/b) = pm a/sqrt{a^2+b^2} #

#tan arctan(a/b) = a/b #

#cot arctan(a/b) = b/a #

#csc arctan(a/b) = pm sqrt{a^2+b^2}/a#

#sec arctan(a/b) = pm sqrt{a^2+b^2}/b#

If they have a square root they get a #pm#.