If sinx=-4/5 what is the exact value of cos2x? (quadrant 3)

Answer 1

It's a 3-4-5 triangle, or rather a -3,-4, 5 right triangle in quadrant III

sinx = -4/5, cosx =-3/5

cos2x = cos^2(x) - sin^2(x) = 9/25 - 16/25 = -7/25

or you could calculate the angle

sinx= -4/5 means x = sin^-1(-4/5) = about 180+53.14 degrees = 233.14 degrees in quadrant III

cos(2 times 233.14) = cos(466.14) = cos(466.14-360) = cos(106.14) = - -0.278

7/25 = 28/100 = 0.28

this all assumes that angle x is in quadrant III, but does it say that or does it say that 2x is in quadrant III?

if 2x is in quadrant III, x might be in quadrant IV and = -53.14 or 360-53.14 = 306.28

2x then would be 612.56 or 612.56-360 = 252.56 which is in quadrant III

cos256.56 = -0.3

but that's not the "exact" value which is slightly greater than -0.3

cos2x = cos^2x - sin^2x is still -7/25 = -28/100 = -0.28 which is slightly > -0.3

same answer no matter how you interpret the problem

Answer 2

We can use the double angle formula for cosine

cos(2x) = cos²(x) - sin²(x)

For the third quadrant, cos(x) = -3/5 ( 3-4-5 triangle)

cos(2x) = (-3/5)² - (-4/5)² = 9/25 - 16/25 = -7/25