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Geometry Level 2

Given that cos ⁡ ( 3 θ ) = a cos ⁡ 3 θ − b cos ⁡ θ \cos (3 \theta) = a \cos ^3 \theta - b \cos \theta is a trigonometric identity, what is the value of a + b a + b ?


The answer is 7.

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Arron Kau by Arron Kau

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1 solution

Kushagra Sahni
Mar 28, 2014

LET THE ANGLE THETA EQUAL TO ANGLE A BECAUSE I CAN'T FIND A WAY TO TYPE THETA cos(3A) =cos(2A+A)= cos2AcosA -- sin2AsinA = (cos^2A-sin^2A)cosA -- (2sinAcosA)sinA=cos^3A -- sin^2AcosA --2sin^2AcosA= cos^3A -- 3sin^2AcosA=cosA(cos^2A -- 3sin^2A)= cosA(cos^2A -- 3(1--cos^2A))= cosA(cos^2A--3+3cos^2A)= cosA(4cos^2A--3)= 4cos^3A--3cosA THEREFORE, a=4, b=3, and a+b=4+3=7 ANS--- 7

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