solve the equation cos^x=sin^x+1

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Supposing that you want to solve the equation cos x = sin x + 1, we'll start by saying that this equation is linear and we'll re-write it moving the function sin x, to the left.

cos x - sin x = 1

We'll solve the equation using a helping angle:

We'll write the coefficient of sin x, namely 1, as the tangent function of pi/4 angle.

We'll re-write the equation:

cos x - tan(pi/4)*sin x = 1

But tan pi/4 = sin pi/4/cos pi/4

cos x - (sin pi/4/cos pi/4)*sin x = 1

cos x*cos pi/4 - sin x*sin pi/4 = cos pi/4

cos (pi/4 + x) = sqrt2/2

pi/4 + x = +/-arccos (sqrt2/2) + 2*k*pi

pi/4 + x = +/-pi/4 + 2kpi

We'll solve the 1st case:

pi/4 + x = pi/4 + 2kpi

x = 2kpi

We'll solve the 2nd case:

pi/4 +x = -pi/4 + 2kpi

x = 2kpi - pi/2

The solutions of the linear trigonometric equation are:{2kpi}U{2kpi - pi/2}.

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