Trigonometric Functions

Trigonometric Functions Problem Solving


θ \theta is an acute angle such that tan(θ)=13\tan (\theta) = \frac{1}{3}. What is the value of 1010(sinθ+cosθ)10 \sqrt{10}\cdot\left(\sin \theta + \cos \theta \right)?

Let mm and MM be the minimum and maximum values of the domain of f(x)=sin1(x2360)f(x) = \sin^{-1}(x^2 - 360), respectively. What is the value of MmM - m?

Details and assumptions

sin1x\sin^{-1} x denotes the functional inverse of sinx\sin x not the reciprocal 1sinx.\frac{1}{\sin x}.

Let OO be the origin and PP be a point in the fourth quadrant on the x-y plane. Let 270<θ<360270 ^\circ < \theta < 360^\circ be an angle formed by OPOP with the positive x-axis. Similarly QQ is a point in the fourth quadrant on the x-y plane where OQ=OP|OQ| = |OP| and the angle formed by OQOQ and the x-axis is 7θ7\theta. For what value of θ\theta does the segments OPOP and OQOQ coincide?

Let θ\theta be the angle between the x-axis and the line connecting the origin O(0,0)O (0,0) and the point P(8,15)P (-8,-15), where 180<θ<270180^\circ < \theta < 270^\circ. Given that sinθ+cosθ+tanθ=ab\sin \theta + \cos \theta + \tan \theta = \frac{a}{b}, where aa and bb are coprime positive integers. What is the value of a+ba+b?


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