Answer:
To determine the value of P, let's start by solving the equation 2cos(2P) = -√3. Since we know that cos(π/6) = √3/2, we can write:
2cos(2P) = 2cos(π/6)
2P = π/6
P = π/12
Next, let's consider the condition tan(2P) < 0. The tangent function is negative in the second and fourth quadrants. Since 2P = π/6, which is in the first quadrant, we can ignore the fourth quadrant. Thus, the solution lies in the second quadrant.
In the second quadrant, the angle is π + π/6 = 7π/6.
Now, let's find all the values of P in the interval [180°, 360°]:
P = π/12 (30°) in the first quadrant
P = 7π/6 (210°) in the second quadrant
Finally, we can determine the value of 3sin(P - 23.5°):
P - 23.5° = π/12 - 23.5°
≈ 15.651°
3sin(15.651°) ≈ 0.802
Therefore, the value of 3sin(P - 23.5°), correct to 3 decimal places, is approximately 0.802.
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