From AutoMATHic

Geometry Level 4

Let the point ( x , y ) (x,y) be a point along the unit circle x 2 + y 2 = 1 x^2+y^2=1 in the first quadrant and θ \theta be the angle measured counterclockwise from the positive x x -axis such that θ = cos ⁡ − 1 ( 4 x + 3 y 5 ) \theta = \cos^{-1} \left( \dfrac{4x+3y}5 \right) .

If tan ⁡ θ = a b \tan \theta = \dfrac ab , where a a and b b are coprime positive integers, find the value of a 2 + b 2 a^2+b^2 .


The answer is 10.

Jun Arro Estrella by Jun Arro Estrella , 23, Philippines

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1 solution

Pulkit Gupta
Dec 23, 2015

This is not a solution but a hint :

For a circle x 2 + y 2 \large x^2+y^2 = r 2 \large r^2 , any point lying on its circle can be represented in its polar form as r cos ⁡ θ \large r \cos \theta & r sin ⁡ θ \large r \sin \theta ( where θ \large \theta is the angle measured from the x axis ).

Put the above values of x & y in the expression for θ \large \theta in the question above. Simplify.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Thanks for the hint :)

Jun Arro Estrella - 5 years, 5 months ago

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