SOLUTION: Assume cos(x) + 2 sin(x) = 11/5 and 4 cos(x) + sin(x) = 16/5 Find the exact (numeric) value of sec(x). Find the exact (numeric) value of tan(x).

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Question 856137: Assume
cos(x) + 2 sin(x) =
11/5

and
4 cos(x) + sin(x) =
16/5

Find the exact (numeric) value of sec(x).
Find the exact (numeric) value of tan(x).
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
you basically have 2 equations that need to be solved simultaneously so you need to follow the rules of solving equations simultaneously and you should arrive at your answer.

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: