secx = 4 in Q2 find sin2x,cos2x,tan2x
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my appologies here is the exact question copied:
Find sin(2x), cos(2x), and tan(2x) from the given information.
sec(x) = 4, x in Quadrant IV
To find sin(2x), cos(2x), and tan(2x) given that sec(x) = 4 in the second quadrant (Q2), we first need to find the value of x.
Since sec(x) is equal to 4, we know that cos(x) = 1/sec(x) = 1/4. Since x is in the second quadrant (Q2), cos(x) will be negative. Therefore, cos(x) = -1/4.
To find sin(x), we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1. Plugging in the known values, we have sin^2(x) + (-1/4)^2 = 1. Simplifying this equation, we get sin^2(x) + 1/16 = 1. Rearranging the equation, we find sin^2(x) = 1 - 1/16 = 15/16. Taking the square root of both sides, we get sin(x) = sqrt(15)/4.
Now that we have the values of sin(x) and cos(x), we can use double angle identities to find sin(2x), cos(2x), and tan(2x).
The double angle identities are:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x)
tan(2x) = (2tan(x))/(1 - tan^2(x))
Plugging in the values, we get:
sin(2x) = 2 * (sqrt(15)/4) * (-1/4) = -sqrt(15)/8
cos(2x) = (-1/4)^2 - (sqrt(15)/4)^2 = 1/16 - 15/16 = -14/16 = -7/8
tan(2x) = (2 * (-1/4))/(1 - (-1/4)^2) = (-1/2)/(15/16) = -8/15
Therefore, in the second quadrant (Q2), sin(2x) = -sqrt(15)/8, cos(2x) = -7/8, and tan(2x) = -8/15.