44) Angle x lies in the third quadrant, and tan x =
9
40
. Determine an exact value for cos(2x).
Since angle x lies in the third quadrant, we know that both the sine and cosine of x will be negative.
Given that tan x = 9/40, we can find the values of sine x and cosine x using the following identities:
tan x = sin x / cos x
9/40 = sin x / cos x
Using the Pythagorean identity sin^2 x + cos^2 x = 1, we can rewrite the equation as:
(40^2)(sin^2 x) + (40^2)(cos^2 x) = (9^2)(sin^2 x)
1600(sin^2 x) + 1600(cos^2 x) = 81(sin^2 x)
1521(sin^2 x) = 1600(cos^2 x)
cos^2 x = (1521/1600)(sin^2 x)
Now, we can find the value of cos^2 x:
cos^2 x = (1521/1600)(1 - cos^2 x)
1600(cos^2 x) = 1521(1 - cos^2 x)
1600(cos^2 x) = 1521 - 1521(cos^2 x)
1521(cos^2 x) = 1521
cos^2 x = 1521/1521
cos^2 x = 1
cos x = √1
cos x = 1 (since x lies in the third quadrant and we know that cos x is negative)
Now, we can find cos(2x) using the double angle cosine identity:
cos(2x) = 2(cos^2 x) - 1
cos(2x) = 2(1) - 1
cos(2x) = 2 - 1
cos(2x) = 1
Therefore, the exact value for cos(2x) is 1.