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Geometry

# Graphs of Trigonometric Functions - Problem Solving

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

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

Consider the function $f(x)=a \cos bx+c,$ where $$a>0$$ and $$b>0$$. If the period of $$f(x)$$ is $$\frac{2}{5}\pi$$, the maximum value of $$f(x)$$ is $$13$$, and $$f(\pi)=-11$$, what is the value of $$abc$$?

Consider the function $f(x)=a|\cos bx|+c,$ where $$a>0$$ and $$b>0$$. If the period of $$f(x)$$ is $$\frac{\pi}{7}$$, the maximum value is $$27$$, and $$f\left(\frac{\pi}{21}\right)=18$$, what is the value of $$a+b+c$$?

Consider the function $f(x)=a \sin \left(\frac{x}{b}-\frac{\pi}{3}\right)-c,$ where $$a>0$$ and $$b>0$$. If the period of $$f(x)$$ is $$34\pi$$, the maximum value is $$21$$, and $$f\left(\frac{17}{6}\pi\right)=0$$, what is the value of $$a+b-c$$?

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