x and y represent two angles in standard position. x has its terminal arm in the first quadrant and y has its terminal arm in the third quadrant. If cosx=5/13 and cosy = -5/13 then find the value of
2sinx+2siny+2cosx-2tanx+tany+2cosy
if cosx = 5/13, (recognize the 5,12,13 triangle ?)
sinx = 12/13
tanx = 12/5
if cosy = -5/13 and y is in III
siny = -12/13
tany = 12/5
2sinx+2siny+2cosx-2tanx+tany+2cosy
= 2(12/13) + 2(-12/13) + 2(5/13) -2(12/5) + (12/5) + 2(-5/13)
= - 12/5 or -2.4
To find the value of the expression 2sinx + 2siny + 2cosx - 2tanx + tany + 2cosy, we first need to identify the values of sinx, siny, cosx, cosy, and tanx.
Given that x has its terminal arm in the first quadrant, cosx = 5/13. We can use the Pythagorean Identity for cosine, sin^2x + cos^2x = 1, to find the value of sinx.
sin^2x + cos^2x = 1
sin^2x + (5/13)^2 = 1
sin^2x + 25/169 = 1
sin^2x = 144/169
sinx = ±12/13
However, since x is in the first quadrant, we can take the positive value, sinx = 12/13.
Moving on to y, since it has its terminal arm in the third quadrant, cosy = -5/13. Again, we can use the Pythagorean Identity to find the value of siny.
sin^2y + cos^2y = 1
sin^2y + (-5/13)^2 = 1
sin^2y + 25/169 = 1
sin^2y = 144/169
siny = ±12/13
Since y is in the third quadrant, siny is negative, siny = -12/13.
Now, we can substitute the values of sinx, siny, cosx, cosy, and tanx into the expression:
2sinx + 2siny + 2cosx - 2tanx + tany + 2cosy
= 2(12/13) + 2(-12/13) + 2(5/13) - 2(tanx) + (-12/13) + 2(-5/13)
Since tanx = sinx/cosx, we can substitute sinx and cosx:
= 2(12/13) + 2(-12/13) + 2(5/13) - 2(sin x/cos x) + (-12/13) + 2(-5/13)
Finally, we can simplify the expression:
= 24/13 - 24/13 + 10/13 - 2(sin x/cos x) - 12/13 - 10/13
= 0 - 2(sin x/cos x)
= -2tanx
Therefore, the value of 2sinx + 2siny + 2cosx - 2tanx + tany + 2cosy is equal to -2tanx.