assignment 7.5: solving trigonometric equations\nscore: 60/100 answered: 7/10\nquestion 8\nsolve ( 6 sin (2…

Explanation:

Step1: Use double - angle formula

Recall the double - angle formula (\sin(2\beta)=2\sin\beta\cos\beta). The equation (6\sin(2\beta)-8\sin\beta = 0) becomes (6\times(2\sin\beta\cos\beta)-8\sin\beta=0). Simplify it to (12\sin\beta\cos\beta - 8\sin\beta=0). Factor out (4\sin\beta): (4\sin\beta(3\cos\beta - 2)=0).

Step2: Solve (\sin\beta = 0)

If (\sin\beta=0), then (\beta = k\pi), (k\in\mathbb{Z}). Since (0\leq\beta<2\pi), when (k = 0), (\beta=0); when (k = 1), (\beta=\pi).

Step3: Solve (3\cos\beta-2 = 0)

If (3\cos\beta-2 = 0), then (\cos\beta=\frac{2}{3}). Using (\beta=\cos^{-1}(x)), (\beta=\cos^{-1}(\frac{2}{3})\approx0.84) or (\beta = 2\pi-\cos^{-1}(\frac{2}{3})\approx5.44) (because (\cos(2\pi - x)=\cos x) and for (y = \cos x), (x\in[0,2\pi]), if (\cos x = a), (x=\cos^{-1}(a)) or (x = 2\pi-\cos^{-1}(a)) when (a\in[- 1,1]))

Answer:

(0,\pi,0.84,5.44)