Geometry

Trigonometric Functions

Trigonometric Functions: Level 3 Challenges

         

Find the value of sin21+sin22sin23++sin288sin289+sin290-\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}

[log10(tan(1))]×[log10(tan(2))]×[log10(tan(3))]××[log10(tan(88))]×[log10(tan(89))]= ? \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 xyxy-plane, and its initial position is (1,0).(1,0).

Let SkS_k denote the circle with radius kk centered around the origin. Starting from (1,0)(1,0), the ant walks 1 unit counter-clockwise on S1.S_{1}. Then, it walks directly (radially outward) to S2,S_2, on which it will walk 2 units counter-clockwise. Then, it will walk directly to S3S_3 and walk 3 units counter-clockwise, and so, with the ant walking kk units on Sk.S_k. (See the image above.)

When the ant crosses the positive xx-axis for the first time since it left (1,0)(1,0), it is on SnS_{n}. What is nn?

Find the value of

cos475+sin475+3sin275cos275cos675+sin675+4sin275cos275.\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)=tan1(x2+1x2+3)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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