answer:To solve this problem, we need to use the fact that if the graph is symmetrical about the point left(frac{4pi}{3}, 0right), then the value of the function at this point must be zero. Therefore, we have: 3cosleft(2 cdot frac{4pi}{3} + varphiright) = 0 This simplifies to: cosleft(frac{8pi}{3} + varphiright) = 0 From the properties of the cosine function, we know that cosleft(thetaright) = 0 when theta = (2k+1)frac{pi}{2} for any integer k. Therefore, we can write: frac{8pi}{3} + varphi = (2k+1)frac{pi}{2} Solving for varphi, we have: varphi = (2k+1)frac{pi}{2} - frac{8pi}{3} varphi = kpi + frac{pi}{2} - frac{8pi}{3} varphi = kpi - frac{13pi}{6} To find the minimum value of |varphi|, we must consider the values of k that make varphi as close to zero as possible. If k=2, we get: varphi = 2pi - frac{13pi}{6} = frac{12pi}{6} - frac{13pi}{6} = -frac{pi}{6} And hence: |varphi| = left|-frac{pi}{6}right| = frac{pi}{6} This is indeed the minimum value because any other integer value for k would result in a larger absolute value for varphi. Therefore, the answer is A: boxed{frac{pi}{6}}.

answer:1. Consider that the rays (OA) and (OB) form a right angle, i.e., ( angle AOB = 90^circ ). 2. Petya drew two rays (OC) and (OD) inside this right angle, forming an angle of (10^circ). Denote the angles created between the rays as follows: - ( angle AOC = alpha ) - ( angle COD = 10^circ ) - ( angle DOB = beta ) 3. We are given that the sum of the largest and smallest acute angles between any drawn rays is (85^circ). First, let's identify the possible angles: - ( angle AOC = alpha ) - ( angle COD = 10^circ ) - ( angle DOB = beta ) - Additionally, ( angle AOD = alpha + 10^circ ) - And ( angle COB = beta + 10^circ ) 4. Assuming the smallest angle is (10^circ) (i.e., (angle COD)), we need to find the largest angle that could potentially be the sum with (10^circ) giving (85^circ). Since ( alpha + beta + 10^circ ) accounts for the entire right angle: [ alpha + 10^circ + beta = 90^circ ] Rearrange to find the relationship between (alpha) and (beta): [ alpha + beta = 80^circ ] 5. Next, the largest angle must include (10^circ) to satisfy the sum of (85^circ): [ text{Largest angle} + 10^circ = 85^circ ] Solving for the largest angle: [ text{Largest angle} = 85^circ - 10^circ = 75^circ ] 6. Now, finding (beta): [ beta + 10^circ = 75^circ implies beta = 65^circ ] 7. From the sum of angles: [ alpha + beta = 80^circ implies alpha + 65^circ = 80^circ implies alpha = 15^circ ] 8. Hence, the three angles dividing the right angle are: - ( alpha = 15^circ ) - ( beta = 65^circ ) - ( 10^circ ) Conclusion: [ boxed{65^circ, 15^circ, 10^circ} ]

answer:Both m and n can be presented as 13k + 3 due to their congruence relation x equiv 3 pmod{13}. - **Finding m:** 13k + 3 geq 100 implies 13k geq 97 implies k geq frac{97}{13}, so k = 8 (smallest integer satisfying condition). Therefore, m = 13(8) + 3 = 107. - **Finding n:** 13l + 3 geq 1000 implies 13l geq 997 implies l geq frac{997}{13}, so l = 77 (smallest integer satisfying condition). Therefore, n = 13(77) + 3 = 1003. - **Calculating n - m:** n - m = 1003 - 107 = boxed{896}.

answer:Solution: Since frac {-2i}{1+i} = frac {-2i(1-i)}{(1+i)(1-i)}=-i(1-i)=-1-i, Therefore, the coordinates of the point corresponding to the complex number frac {-2i}{1+i} are (-1, -1). Hence, the answer is: boxed{(-1, -1)}. This is obtained by directly using the algebraic form of complex number multiplication and division to simplify the expression. This question tests the algebraic form of complex number multiplication and division, and examines the algebraic representation of complex numbers and their geometric meaning. It is a basic question.