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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.

Using Triple Angle Formula

         

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

\[ \cos 3 \theta = 4 \cos \theta + 3 ? \]

If provided that \[\sin x+\cos x=\frac{1}{6},\] what is the value of \(\sin 3x-\cos 3x ?\)

Suppose that \(\displaystyle \cos \theta=\frac{\sqrt{2}}{4}\) for \(\displaystyle 0 < \theta < \frac{\pi}{2}.\) Then \(\sin 3\theta+\cos 3\theta\) can be expressed as \[\sin 3\theta+\cos 3\theta=\frac{a\sqrt{2}+b\sqrt{14}}{8}\] for rational numbers \(a\) and \(b.\) What is the value of \(3a+8b ?\)

If provided that \[\sin x+\cos x=\frac{\sqrt{3}}{5},\] where \(\sin x > \cos x,\) what is the value of \(\sin 3x+\cos 3x ?\)

Suppose that \(\sin 3\theta=\sin 2\theta\) for \(\displaystyle 0 < \theta < \frac{\pi}{2}.\) Then what is the value of \(20\cos^2 \theta-10\cos \theta ?\)

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