What is sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x) if sin(2x)=5/13 and x is in between 0 and 45 degrees?

Question

The answers have to be exact.

Answer 1

If sin(2x) = 5/13 then cos(2x) = 12/13

[Note that we get cos(2x) either by drawing the 5-12-13 right triangle

or from the identity cos²(2x) = 1 – sin²(2x) = 1 – 25/169 = 144/169]

We know that our angle is in the first quadrant, so all six trig functions will have positive values.

We can use half-angle formulas to find sin(x) and cos(x)

sin(x) = √[(1 – cos(2x))/2] = √[(1 – 12/13)/2] = √[(1/13)/2] = 1/√26

cos(x) = √[(1 + cos(2x))/2] = √[(1 + 12/13)/2] = √[(25/13)/2] = 5/√26

Check: sin²(x) + cos²(x) = 1/26 + 25/26 = 1

tan(x) = sin(x)/cos(x) = (1/√26)/(5/√26) = 1/5

csc(x) = 1/sin(x) = √26

sec(x) = 1/cos(x) = (√26)/5

cot(x) = 1/tan(x) = 5