How do you find the exact value of cos(36°) using the sum and difference, double angle or half-angle formulas?
Step 1: Express cos(36°) as a simpler form
Given: cos(36°)
Let A=18°
5A=90°2A+3A=90°2A=90°-3A
Taking sin on both sides
sin(2A)=sin(90°-3A) [sin(90°-A)=cos(A)]
sin(2A)=cos(3A)
2sin(A)cos(A)=4cos3(A)-3cos(A) [sin(2x)=2sinxcosx,cos(3x)=4cos3x-3cosx]
Step 2: Simplify the equation
2sin(A)=4cos2(A)-32sin(A)=41-sin2(A)-32sin(A)=4-4sin2(A)-34sin2(A)+2sin(A)-1=0sin(A)=-2±22-4×(-1)×42×4sin(A)=-2±4+168sin(A)=-2±258sin(18°)=-1±54
Step 3: Solve for the value of cos(36°)
cos(36°)=cos(2×18°) [cos2x=1-2sin2x]
=1-2sin2(18°)=1-2×-1+542=1-2×5+1-2516=1-3-54cos(36°)=1+54
Hence, cos(36°)=1+54 .
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