answer:The opposite of a number is defined as the number that, when added to the original number, results in zero. Let's denote the opposite of -1009 as x. Therefore, we have: [ -1009 + x = 0 ] Solving for x, we get: [ x = 1009 ] Thus, the opposite of -1009 is 1009. Therefore, the correct answer is boxed{C}.

answer:The general form of a sine function is y = sin(Bx - C). The phase shift can be calculated using the formula: [ text{Phase Shift} = frac{C}{B} ] where C is the phase shift constant and B is the coefficient of x inside the sine function. For the function y = sin(5x - frac{3pi}{2}), the values are B = 5 and C = frac{3pi}{2}. Plugging these into the formula gives: [ text{Phase Shift} = frac{frac{3pi}{2}}{5} = frac{3pi}{10} ] The positive value indicates a shift to the right. Therefore, the phase shift of the graph of y = sin(5x - frac{3pi}{2}) is boxed{frac{3pi}{10}}.

answer:Let's call the unknown number "x". We have the equation: (√97 + √486) / √x = 4.340259786868312 First, let's square both sides of the equation to eliminate the square root of x: [(√97 + √486) / √x]^2 = 4.340259786868312^2 (√97 + √486)^2 / x = 18.8376 (approximately) Now, let's simplify the left side of the equation by squaring the sum of the square roots: (√97 + √486)^2 = (√97)^2 + 2(√97)(√486) + (√486)^2 = 97 + 2√(97*486) + 486 Now, let's calculate the value of 2√(97*486): 2√(97*486) = 2√(47022) ≈ 2*216.842 = 433.684 (approximately) So, the equation becomes: (97 + 433.684 + 486) / x = 18.8376 Now, let's add the numbers in the numerator: 1016.684 / x = 18.8376 Now, to find x, we divide 1016.684 by 18.8376: x = 1016.684 / 18.8376 ≈ 53.97 So, the number x is approximately boxed{53.97} .

answer:Since asin A+bsin B-csin C= frac{6sqrt{7}}{7}asin Bsin C, We can rewrite this using the Law of Sines and Cosines as: a^{2}+b^{2}-c^{2}= frac{6sqrt{7}}{7}absin C=2abcos C, This implies that frac{6sqrt{7}}{7}sin C=2cos C, which gives us tan C= frac{2sqrt{7}}{6}, and cos C=sqrt{frac{1}{1+tan^{2}C}}=frac{3}{4}, Given that a=3 and b=2, We can use the cosine rule to get cos C=frac{a^{2}+b^{2}-c^{2}}{2ab}=frac{9+4-c^{2}}{2times3times2}=frac{3}{4}, Solving for c, we get c=2. Hence, the answer is boxed{2}. This problem primarily tests the application of the sine law, cosine law, and basic trigonometric identities in solving triangles. It requires computational skills and the ability to transform equations. The problem is of moderate difficulty.