Solving a Trigonometric Inequality
To solve this inequality, we can start by rewriting it as sin(π/3 - 2x) cos (π/3 - 2x) ≥ -√3/4.
Next, we can use the trigonometric identity sin(a) cos(a) = (1/2) sin(2a) to simplify the left side of the inequality. This gives us (1/2) sin(2(π/3-2x)) ≥ -√3/4.
Now, we can further simplify by using the double angle identity sin(2θ) = 2sin(θ)cos(θ). Thus, we have sin(π/3 - 2x) ≥ -√3/8.
We can then solve for x by taking the inverse sine of both sides and using the unit circle to find all possible solutions. In this case, we get x ≤ 1/4 or x ≥ 5/12.
However, we need to be careful as we could have extraneous solutions. To check for this, we can substitute each solution into the original inequality to see if it still holds true.
In conclusion, the solutions to the inequality sin(3,14/3-2x) cos (3,14/3-2x) ≥ - √3/4 are x ≤ 1/4 or x ≥ 5/12.
Explanation:
The inequality given in the query is an example of a trigonometric inequality. The steps to solve this inequality involve simplifying, using trigonometric identities, and checking for possible extraneous solutions. This problem requires a deep understanding of trigonometry and a systematic approach to solving inequalities.
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