Given that tan x= -3/4 and x is in the second quadrant. Find sin 2x, cos 2x, and tan 2x (Exact, no decimals).
Please show step-by-step, thank you!
To find sin 2x, cos 2x, and tan 2x, we can use the identities involving double angles.
Step 1: Find sin x, cos x, and tan x.
Since tan x is given as -3/4, we can use the Pythagorean identity to find sin x and cos x.
cos x = sqrt(1 / (1 + tan^2 x))
= sqrt(1 / (1 + (-3/4)^2))
= sqrt(1 / (1 + 9/16))
= sqrt(1 / (25/16))
= sqrt(16/25)
= 4/5
sin x = tan x * cos x
= (-3/4) * (4/5)
= -12/20
= -3/5
Step 2: Use double angle identities.
sin 2x = 2 * sin x * cos x
cos 2x = cos^2 x - sin^2 x
tan 2x = (2 * tan x) / (1 - tan^2 x)
Plugging in the values we found in step 1:
sin 2x = 2 * (-3/5) * (4/5)
= -24/25
cos 2x = (4/5)^2 - (-3/5)^2
= 16/25 - 9/25
= 7/25
tan 2x = (2 * (-3/4)) / (1 - (-3/4)^2)
= -6/4 / (1 - 9/16)
= -6/4 / (7/16)
= -6/4 * (16/7)
= -24/7
Therefore, sin 2x = -24/25, cos 2x = 7/25, and tan 2x = -24/7.
construct a triangle in quad II with x=-4 and y = 3
since tanØ = y/x or opposite/adjacent
so r^2 = (-4)^2 + 3^2
r = 5
sinx = 3/5 and cosx = -4/5
you have to know that
sin 2x = 2sinxcosx = 2(3/5)(-4/5) = -24/25
cos 2x = cos^2 x - sin^2 x = 16/25 - 9/25 = 7/25
tan 2x = sin 2x/ cos 2x = (-24/25) / (7/25) = -24/7