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Trigonometric Equations

Equations with trigonometry in them can look scary, but that's nothing that a quick little substitution can't fix.

R method

         

In the interval \( [0, 20 \pi ] \), how many solutions are there to

\[ \sin \theta + \cos \theta = 1? \]

What is the maximum value of the function \[f(x)=2\sin \left(x+\frac{\pi}{3}\right)+\sqrt{3}\cos x+2 ?\]

The sum of all the \(\theta\)'s in the interval \(12\pi < \theta < 14\pi\) such that the quadratic equation for \(x\) \[x^2-2(\sin \theta+\cos \theta)x+\cos 2\theta=0\] has a repeated root is \(\frac{a}{2}\pi\). What is the value of \(a?\)

If provided that \[\sqrt{3}\cos \theta+\sin \theta=\frac{\sqrt{3}}{2}\] and \(\displaystyle \frac{\pi}{2} < \theta < \pi,\) what is the value of \[\sqrt{3}\sin \theta-\cos \theta ?\]

What are the maximum and minimum values of the function \[y=\sqrt{2}\sin x-\cos x ?\]

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