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Trigonometric Functions

SOH-CAH-TOA! Have you heard the call of the trig function? Use them to navigate the trickier sides (and angles) of geometric shapes.

Problem Solving

\( \theta\) is an acute angle such that \(\tan (\theta) = \frac{1}{3}\). What is the value of \(10 \sqrt{10}\cdot\left(\sin \theta + \cos \theta \right)\)?

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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 the 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 \(a+b\)?

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