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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.

Solve by Factoring

         

In the domain \( [0, 2 \pi ] \), how many solutions are there to

\[ 6 \cos^2 \theta - 10 \cos \theta + 3 \sin^2 \theta = 0 ? \]

Which of the following are the solutions to the equation

\[ 6 \tan^2 \theta - 17 \tan \theta + 2 \sec^2 \theta = 0 ? \]

If \(2\pi \leq x < 4\pi\), the sum of all the \(x\)'s that satisfy \[\tan x+ \frac{1}{\tan x}-2=0\] can be expressed as \(a\pi\). What is the value of \(2a?\)

The two roots of the quadratic equation \[2x^2-x+k=0\] are \(\sin \theta\) and \(\cos \theta\). What is \[16(\sin^3 \theta+\cos^3 \theta)? \]

In the domain \( [0^\circ , 1000^\circ ] \), how many solutions are there to \( 4 \sin^2 \theta - 3 \sin \theta - 1 = 0 \)?

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