Sum and Difference Trigonometric Formulas
Given that \(\sin^2 \theta = \frac{1}{3}\), what is the value of \(72 \cos(2 \theta)\)?
It is given that \(\sin(\theta) = \frac{1}{\sqrt{37}}\). If \( \cos ( 2 \theta ) = \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?
\(x\) is an angle measured in radians and satisfies \(\frac{-\pi}{2} < x < \frac{\pi}{2}\). It is given that \(\sin(x) = \frac{8}{17}\). If \(\sin(2x)\) can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are positive, coprime integers, what is the value of \(a+b\)?
The double angle identity states that
\[ \cos (2 \theta) = N \cos^2 \theta - M, \]
where \(N\) and \(M\) are real numbers. What is the value of \( N + M \)?
The double angle identity states that
\[ \sin ( 2 \theta ) = N \sin \theta \cos \theta + M, \]
where \(N\) and \(M\) are real numbers. What is the value of \(N + M \)?