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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 $$10 \sqrt{10}\cdot\left(\sin \theta + \cos \theta \right)$$?



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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