3 Calculate the value of sin 2A if cos A = 24/25 and A is in the first quadrant. 3.4 Solve for x if: 3 cos x tan x = tan x Where, 0° ≤ x ≤ 360° 3.5 Simplify: cos(315) sin(210) tan(225) / sin(-45) cos(240)

Here are the solutions to your questions: 3.3 Calculate the value of 2A if A = (24)/(25) and A is in the first quadrant. Step 1: Find A. Since A is in the first quadrant, A > 0. We use the Pythagorean identity ^2 A + ^2 A = 1. ^2 A + ((24)/(25))^2 = 1 ^2 A + (576)/(625) = 1 ^2 A = 1 - (576)/(625) = (625 - 576)/(625) = (49)/(625) A = sqrt((49)/(625)) = (7)/(25) Step 2: Calculate 2A using the double angle identity 2A = 2 A A. 2A = 2 ((7)/(25)) ((24)/(25)) 2A = 2 × (168)/(625) 2A = (336)/(625) The value of 2A is (336)/(625). 3.4 Solve for x if 3 x x = x, where 0^ x 360^. Step 1: Rearrange the equation and factor. 3 x x - x = 0 x (3 x - 1) = 0 Step 2: Solve for each factor. Case 1: x = 0 This occurs when x = 0. x = 0^, 180^, 360^ (Note: x is undefined at 90^ and 270^, but these are not solutions for x = 0.) Case 2: 3 x - 1 = 0 3 x = 1 x = (1)/(3) The reference angle is = ((1)/(3)) ≈ 70.53^. Since x is positive, x is in Quadrant I or Quadrant IV. In Quadrant I: x_1 ≈ 70.53^ In Quadrant IV: x_2 = 360^ - ≈ 360^ - 70.53^ ≈ 289.47^ Step 3: Combine all solutions within the given range. The solutions for x are approximately 0^, 70.53^, 180^, 289.47^, 360^. 3.5 Simplify: ((315^) (210^) (225^))/((-45^) (240^)) Step 1: Evaluate each trigonometric function using reference angles and quadrant rules. (315^) = (360^ - 45^) = (45^) = sqrt(2)2 (210^) = (180^ + 30^) = -(30^) = -(1)/(2) (225^) = (180^ + 45^) = (45^) = 1 (-45^) = -(45^) = -sqrt(2)2 (240^) = (180^ + 60^) = -(60^) = -(1)/(2) Step 2: Substitute the values into the expression and simplify. (sqrt(2)2) (-(1)/(2)) (1)(-sqrt(2)2) (-(1)/(2)) = -sqrt(2)4sqrt(2)4 = -1 The simplified value is -1. Last free one today — make it count tomorrow, or type /upgrade for unlimited.