Trigonometric Functions
\( \theta\) is an acute angle such that \(\tan (\theta) = \frac{1}{3}\). What is the value of \(10 \sqrt{10}\cdot\left(\sin \theta + \cos \theta \right)\)?
Let \(m\) and \(M\) be the minimum and maximum values of the domain of \(f(x) = \sin^{-1}(x^2 - 360)\), respectively. What is the value of \(M - m\)?
Details and assumptions
\(\sin^{-1} x\) denotes the functional inverse of \(\sin x\) not the reciprocal \(\frac{1}{\sin x}.\)
Let \(O\) be the origin and \(P\) be a point in the fourth quadrant on the x-y plane. Let \(270 ^\circ < \theta < 360^\circ \) be an angle formed by \(OP\) with the positive x-axis. Similarly \(Q\) is a point in the fourth quadrant on the x-y plane where \(|OQ| = |OP|\) and the angle formed by \(OQ\) and the x-axis is \(7\theta\). For what value of \(\theta\) does the segments \(OP\) and \(OQ\) coincide?
Let \(\theta\) be the angle between the x-axis and the line connecting the origin \(O (0,0)\) and the point \(P (-8,-15)\), where \(180^\circ < \theta < 270^\circ\). Given that \(\sin \theta + \cos \theta + \tan \theta = \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?