answer:Let's analyze each option step by step: **For option A:** We start with the given formula for cos 3theta: cos 3theta = cos(2theta + theta) Applying the sum formula for cosine, we get: = cos 2theta cos theta - sin theta sin 2theta Substituting the double angle formulas for cos 2theta and sin 2theta, we have: = (2cos^2theta - 1)cos theta - 2sin^2theta cos theta Using the Pythagorean identity sin^2theta = 1 - cos^2theta, we can rewrite the equation as: = (2cos^2theta - 1)cos theta - 2(1 - cos^2theta)cos theta Simplifying, we find: = 4cos^3theta - 3cos theta Thus, option A is correct. **For option B:** Given cos 72^circ = sin 18^circ, we use the triple angle formula for cosine: cos 3 times 18^circ = cos(90^circ - 2 times 18^circ) This simplifies to: 4cos^3 18^circ - 3cos 18^circ = 2sin 18^circ cos 18^circ Since cos 18^circ neq 0, we divide both sides by cos 18^circ and get: 4sin^2 18^circ + 2sin 18^circ - 1 = 0 Solving this quadratic equation for sin 18^circ yields sin 18^circ = frac{sqrt{5} - 1}{4}, which contradicts the statement in option B that cos 72^circ = frac{sqrt{5} - 1}{2}. Therefore, option B is incorrect. **For option C:** Given the equation 4x^3 - 3x - frac{1}{2} = 0, and letting x = cos theta, we find: cos 3theta = frac{1}{2} This implies 3theta = frac{pi}{3}, frac{5pi}{3}, text{ or } frac{7pi}{3}. The roots are thus: cos frac{pi}{9}, cos frac{5pi}{9}, cos frac{7pi}{9} Calculating the sum of their cubes, we get: 4{x}_{1}^{3} + 4{x}_{2}^{3} + 4{x}_{3}^{3} = 3(x_1 + x_2 + x_3) + frac{3}{2} = 3left(cosfrac{pi}{9} + cosfrac{5pi}{9} + cosfrac{7pi}{9}right) + frac{3}{2} This simplifies to frac{3}{2}, so option C is correct. **For option D:** The sum of cosines at intervals of 72^circ can be expressed as: cos theta + cos(theta + 72^circ) + cos(theta + 144^circ) + cos(theta + 216^circ) + cos(theta + 288^circ) This simplifies to: cos theta + 2cos theta sin 18^circ - 2cos theta cos 36^circ Further simplification gives: cos theta (4sin^2 18^circ + 2sin 18^circ - 1) = 0 Thus, option D is correct. Therefore, the correct options are boxed{ACD}.

answer:The graph of y = gleft( frac{x + 2}{3} right) involves: 1. A horizontal stretch by a factor of 3, making the function slower to change horizontally. 2. A shift to the left by 2 units, adjusting how the function moves along the x-axis. Let's redefine the transformed function: - For x geq -8 and x < -1, after stretching and shifting, the function becomes y = -1 - left(frac{x+2}{3}right) = -frac{x+5}{3}. - For x geq -1 and x leq 7, the second part after stretching and shifting is y = sqrt{9 - left((frac{x+2}{3} - 3)^2right)} - 1 = sqrt{9 - left(frac{x-7}{3}right)^2} - 1. - For x > 7 and x leq 10, the last part becomes y = 3*left(frac{x+2}{3} - 3right) = x - 7. Therefore, the new graph of y = gleft( frac{x + 2}{3} right) is as follows: - A linear decline from x = -8 to x = -1, changing at a rate of frac{-1}{3}. - A circular arc centered at x = 7 with radius 3 from x = -1 to x = 7. - A linear increase from x = 7 to x = 10 with a slope of 1. boxed{text{Transposed piece-wise function}}.

answer:For the domain of the function c(x) to cover all real numbers, the denominator -5x^2 + 3x + k must not be zero for any real x. This implies that the quadratic equation [-5x^2 + 3x + k = 0] must not have real roots. The discriminant (b^2 - 4ac) of this quadratic equation needs to be negative: [(3)^2 - 4(-5)(k) < 0,] [9 + 20k < 0,] [20k < -9,] [k < -frac{9}{20}.] Therefore, the set of all values for k is boxed{left(-infty, -frac{9}{20}right)}.

answer:**Analysis** Using the conclusion that for ngeqslant 2, a_n=s_n-s_{n-1}, combined with the given conditions, it is easy to deduce that a_6 > 0, a_7=0, a_8 < 0. Then, analyze each option one by one. This problem tests the formula and properties of the sum of the first n terms of an arithmetic sequence, as well as the maximum value problem of s_n. Skillful application of the formula is the key to solving the problem, and it is important to make reasonable use of the idea of equivalent transformation. **Solution** Let the common difference of the arithmetic sequence {a_n} be d. For option A, since S_5 < S_6=S_7 > S_8, it follows that S_6-S_5=a_6 > 0, S_8-S_7=a_8 < 0, which means a_6+2d < 0, therefore 2d < -a_6 < 0, hence d < 0. So, option A is correct. For option B, S_9-S_5=a_6+a_7+a_8+a_9=2(a_7+a_8)=2(0+a_8)=2a_8 < 0, thus S_9 < S_5. Therefore, option B is incorrect. For option C, since S_6=S_7, it follows that S_7-S_6=a_7=0. So, option C is correct. For option D, since S_5 < S_6=S_7 > S_8, it follows that both S_6 and S_7 are the maximum values of S_n. Therefore, option D is correct. Hence, the incorrect option is boxed{text{B}}.