Monoxdifly
MHB
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The set of real numbers x at the interval [0, 2π ] which satisfy $$2sin^2x\geq3cos2x+3$$ takes the form [a, b] ∪ [c, d]. The result of a + b + c + d is ...
a. 4π
b. 5π
c. 6π
d. 7π
e. 8π
What I've done thus far:
$$2sin^2x\geq3cos2x+3$$
$$2sin^2x\geq3(cos2x+1)$$
$$2sin^2x\geq3(cos^2x-sin^2x+sin^2x+cos^2x)$$
$$2sin^2x\geq3(2cos^2x)$$
$$sin^2x\geq3cos^2x$$
$$\frac{sin^2x}{cos^2x}\geq3$$
$$tan^2x\geq3$$
$$tanx\geq\sqrt3$$
$$tanx\geq tan60°$$ or $$tanx\geq tan240°$$
Since the value of tangent become negative after 90° and 270° respectively, I assume that [a, b] ∪ [c, d] is [60°, 90°] ∪ [240°, 270°] thus a + b + c + d = 660° = $$3\frac23\pi$$, but it isn't in the options. Where did I do wrong? I'm sure I solved the inequation alright and just blundered in determining the intervals, but how should I fix them?
a. 4π
b. 5π
c. 6π
d. 7π
e. 8π
What I've done thus far:
$$2sin^2x\geq3cos2x+3$$
$$2sin^2x\geq3(cos2x+1)$$
$$2sin^2x\geq3(cos^2x-sin^2x+sin^2x+cos^2x)$$
$$2sin^2x\geq3(2cos^2x)$$
$$sin^2x\geq3cos^2x$$
$$\frac{sin^2x}{cos^2x}\geq3$$
$$tan^2x\geq3$$
$$tanx\geq\sqrt3$$
$$tanx\geq tan60°$$ or $$tanx\geq tan240°$$
Since the value of tangent become negative after 90° and 270° respectively, I assume that [a, b] ∪ [c, d] is [60°, 90°] ∪ [240°, 270°] thus a + b + c + d = 660° = $$3\frac23\pi$$, but it isn't in the options. Where did I do wrong? I'm sure I solved the inequation alright and just blundered in determining the intervals, but how should I fix them?