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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.

Level 3

Find the value of \[-\sin^2{1^\circ}+\sin^2{2^\circ }-\sin^2{3^\circ }+\ldots+\sin^2{88^\circ}-\sin^2{89^\circ}+\sin^2{90^\circ}\]

\[ \left [ \log_{10} \left ( \tan \left ( 1^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 2^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 3^\circ \right ) \right ) \right ] \times \cdot \cdot \cdot \\ \times \left [ \log_{10} \left ( \tan \left ( 88^\circ \right ) \right ) \right ] \times \left [ \log_{10} \left ( \tan \left ( 89^\circ \right ) \right ) \right ] = \ ? \]

An ant finds itself trapped in the \(xy\)-plane, and its initial position is \((1,0).\)

Let \(S_k\) denote the circle with radius \(k\) centered around the origin. Starting from \((1,0)\), the ant walks 1 unit counter-clockwise on \(S_{1}.\) Then, it walks directly (radially outward) to \(S_2,\) on which it will walk 2 units counter-clockwise. Then, it will walk directly to \(S_3\) and walk 3 units counter-clockwise, and so, with the ant walking \(k\) units on \(S_k.\) (See the image above.)

When the ant crosses the positive \(x\)-axis for the first since it left \((1,0)\), it is on \(S_{n}\). What is \(n\)?

Find the value of

\[\frac{\cos^4 75^{\circ}+\sin^4 75^{\circ}+3\sin^2 75^{\circ}\cos^2 75^{\circ}}{\cos^6 75^{\circ}+\sin^6 75^{\circ}+4\sin^2 75^{\circ}\cos^2 75^{\circ}}.\]

If the domain of \[f(x)=\tan^{-1} \left( \dfrac{x^2+1}{x^2+\sqrt{3}}\right)\] is the real numbers, find its range.


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