Geometry

Graphs of Trigonometric Functions

Graphs of Trigonometric Functions - Problem Solving

         

Consider the intersection points of the two functions y=cos26xy=k,  (1<k<0) \begin{aligned} y&=\cos 26x \\ y&=k, \; (-1<k<0) \end{aligned} in the domain 0x326π0 \leq x \leq \frac{3}{26}\pi. If the xx-coordinates of those points are α\alpha, β\beta and γ\gamma in an ascending order, the value of α+2β+γ\alpha+2\beta+\gamma can be expressed as abπ\frac{a}{b}\pi, where aa and bb are coprime positive integers. What is the value of a+ba+b?

If the minimum and maximum value of y=sinx110+18y=\left| \sin x-\frac{1}{10}\right|+18 are aa and bb, respectively, what is the value of a+10ba+10b?

Consider the function f(x)=acosbx+c,f(x)=a \cos bx+c, where a>0a>0 and b>0b>0. If the period of f(x)f(x) is 25π\frac{2}{5}\pi, the maximum value of f(x)f(x) is 1313, and f(π)=11f(\pi)=-11, what is the value of abcabc?

Consider the function f(x)=acosbx+c,f(x)=a|\cos bx|+c, where a>0a>0 and b>0b>0. If the period of f(x)f(x) is π7\frac{\pi}{7}, the maximum value is 2727, and f(π21)=18f\left(\frac{\pi}{21}\right)=18, what is the value of a+b+ca+b+c?

Consider the function f(x)=asin(xbπ3)c,f(x)=a \sin \left(\frac{x}{b}-\frac{\pi}{3}\right)-c, where a>0a>0 and b>0b>0. If the period of f(x)f(x) is 34π34\pi, the maximum value is 2121, and f(176π)=0f\left(\frac{17}{6}\pi\right)=0, what is the value of a+bca+b-c?

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