Given cosθ = 3/5 (π ≤ θ ≤ 3π/2 ) and sin x = 24/25 (0 ≤ x ≤ π 2 ), determine the exact value of cos(θ − x).

Answer 1

pi < theta < 3pi/2

180 < theta < 270 degrees

means theta is in quadrant III, but cosines are always negative in quadrant III and would never = +3/5. Maybe you meant -3/5?

if cos(theta) = -3/5, then sin(theta) = -4/5 or 4/5

if sin(x) = 24/25 then cos(x) = 7/25 or -7/25; x is in either quadrant I or II.

cos(theta - x) = cos(theta)cos(x) + sin(theta)sin(x)

= (-3/5)(7/25 or -7/25) + (4/5 or -4/5)(24/25)

= 4 possibilities

= (21+96)/125 = 117/125

or (-21+96)/125 = 75/125 = 3/5

or (21-96)/125 = -75/125 = -3/5

or (-21-96)/125 = -117/125

cos(theta -x) = +3/5 or +117/125 = + 0.936

but ignore the positive solutions, see below

for cos(theta) = -3/5, theta = about 180-53.13= 126.87 degrees but that's not between 180 and 270

so theta = 180+53.13 = about 233.13 degrees

for sin(x) = 24/25, x = about 73.74 degrees or

about 180-73.74 = 106.26 degrees

theta - x = either 233.13 - 73.74 = about 160 degrees cos(theta-x) = -0.936

or 233.13- 106.26= 126.87 degrees cos(126.87) = -0.6

or 306.87 -106.26 = 200.61 degrees cos(200.61)=-.936

cos(theta-x) =- 3/5 or - 0.936 =-117/125