Trigonometric Equations

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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Trigonometric Equations - Problem Solving

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