Trigonometric Equations

Trigonometric Equations - Problem Solving


If \( 2 \cos^2{x} + 47 \cos{x} = 20 \sin^2{x} , \) then what is the value of \( \cos{x} ? \)

In the domain \( [0, 17 ] \), how many real solutions are there to \( \cos (\pi x^2 ) = -1 \)?

Let \(\alpha\), \(\beta\) and \(\gamma\) be the period, maximum value and minimum value, respectively, of the function \[y=\sin^4 6x+\cos^4 6x.\] What is the value of \(\frac{\beta \pi}{\alpha \gamma}\)?

List the positive roots of the trigonometric equation \[\cos^2 x-\sin^2 2x=0\] in ascending order to form a sequence \(\{a_n\}\). What is the value of the constant \(m\) that satisfies \[\displaystyle \sum_{k=1}^{36}a_k=m\pi?\]

What are all possible values of \(\sin\theta + 8\cos\theta\)?


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