Determine the general solution of tan x = 2 sin 2x where cos x < 0.

Here's a detailed breakdown for question 4: 4. Determine the general solution of x = 2 2x where x < 0. Step 1: Rewrite the equation using fundamental trigonometric identities. Recall that x = ( x)/( x) and 2x = 2 x x. Substitute these into the given equation: ( x)/( x) = 2(2 x x) ( x)/( x) = 4 x x Step 2: Rearrange the equation to set it to zero and factor. Multiply both sides by x (note: x ≠ 0 for x to be defined, so we must check solutions where x = 0 separately if they arise from the original equation, but here we are given x < 0, so x ≠ 0 is already implied). x = 4 x ^2 x Move all terms to one side: x - 4 x ^2 x = 0 Factor out x: x (1 - 4^2 x) = 0 Step 3: Solve for each factor being equal to zero. This gives two separate cases: Case 1: x = 0 Case 2: 1 - 4^2 x = 0 Step 4: Solve Case 1 and apply the condition x < 0. For x = 0, the general solution is x = 180^ k, where k Z. Now, we apply the condition x < 0. • If k is an even integer (e.g., k=0, 2, -2, ), then x = 0^, 360^, -360^, . For these values, x = 1, which does not satisfy x < 0. • If k is an odd integer (e.g., k=1, 3, -1, ), then x = 180^, 540^, -180^, . For these values, x = -1, which satisfies x < 0. So, the solutions from Case 1 that satisfy x < 0 are x = 180^ + 360^ k, where k Z. Step 5: Solve Case 2 and apply the condition x < 0. For 1 - 4^2 x = 0: 4^2 x = 1 ^2 x = (1)/(4) Take the square root of both sides: x = ± sqrt((1)/(4)) x = ± (1)/(2) Now, apply the condition x < 0. This means we only consider x = -(1)/(2). For x = -(1)/(2): The reference angle is 60^. Since x is negative, x lies in the second or third quadrants. • In the second quadrant: x = 180^ - 60^ = 120^. • In the third quadrant: x = 180^ + 60^ = 240^. The general solutions for these are: x = 120^ + 360^ k, where k Z. x = 240^ + 360^ k, where k Z. Step 6: Combine all valid general solutions. The general solutions satisfying x < 0 are: x = 180^ + 360^ k x = 120^ + 360^ k x = 240^ + 360^ k where k Z. The general solution is x = 180^ + 360^ k, x = 120^ + 360^ k, x = 240^ + 360^ k, k Z. That's 2 down. 3 left today — send the next one.