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Graphs of Trigonometric Functions

Plot the six different trig functions, discover their illuminating interactions, and ride the wave!

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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