start with:
cos(x) + 2sin(x) = 11/5 (first equation)
4cos(x)+ sin(x) = 16/5 (second equation)
multiply the second equation by 2 to get:
cos(x) + 2sin(x) = 11/5
8cos(x) + 2sin(x) = 32/5
subtract the first equation from the second equation to get:
7cos(x) = 21/5
divide both sides of this equation by 7 to get:
cos(x) = 21/35
since cos(x) = adjacent / hypotenuse, this means that:
side adjacent to x is 21
hypotenuse is 35
since a^2 + b^2 = c^2 by the pythagorean formula, this means that you can replace a with 21 and c with 35 to get:
21^2 + b^2 = 35^2
you can now solve for b to get:
b = side opposite angle x which is equal to 35^2 - 21^2 which is equal to 28.
your triangle has:
side adjacent to angle x is equal to 21.
side opposite angle x is equal to 28.
hypotenuse is equal to 35.
because similar triangles have proportional sides and equal angles, you can also make a similar triangle by dividing all the sides by 7 to get:
side adjacent to angle is equal to 3.
side opposite angle x is equal to 4.
hypotenuse3 is equal to 5.
this is your standard 3,4,5 triangle.
in this triangle:
sin(x) = opposite / hypotenuse = 4/5.
cos(x) = adjacent / hypotenuse = 3/5.
tan(x) = opposite / adjacent = 4/3.
sec(x) = 1/cos(x) = 1/(4/5) = 5/4.
your answer are:
tan(x) = 4/3.
sec(x) = 5/4.
you can confirm your solution is correct in the following manner.
your original equations are:
cos(x) + 2sin(x) = 11/5 (first equation)
4cos(x)+ sin(x) = 16/5 (second equation)
replace cos(x) with 3/5 and replace sin(x) with 4/5 and see if these equations are true.
you get:
3/5 + 8/5 = 11/5 (first equation is true)
12/5 + 4/5 = 16/5 (second equation is true).
looks like you're good to go.
your solutions are:
tan(x) = 4/3.
sec(x) = 5/4.
here's a picture of your triangle:
