Find the exact values of sin (2 theta) and cos (2 theta) if sec theta = -3 with pi/2 < theta < pi

if secθ = -3 then cosθ = -1/3

if cosθ = -1/3 then sinθ = √2 / 3 because the opposite side of the right triangle in the second quadrant that creates the cosθ has an hypotenuse of 3 and an opposite side of √2.

Using the Double Angle Formula for sine:

sin2θ = 2sinθcosθ = 2(√2/3)(-1/3) = -2√2/9

Using the Double Angle Formula for cosine:

cos2θ = 2(cosθ)^2 - 1 = 2(-1/3)^2 - 1 = 2(1/9) - 1 = 2/9 - 1 = -7/9

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