Math CentralQuandaries & Queries


Question from Emanuel, a student:

If sin3x=cos2x, find the value of x when x<0<90'

Hi Emanuel,

Use the double angle expression for the cosine to write $\cos(2x)$ in terms of $\sin(x).$ Write $3x = 2x + x$ and use an expression for $\sin(A + B)$ to write $\sin(2x + x)$in terms of the sine and cosine of $x$ and $2x.$ Use the double angle expression for $\sin(2x)$ and $\cos(2x)$ to write $\sin(3x)$ in terms of $\sin(x)$ and $\cos(x).$Using these expressions you should now be able to write the equation $\sin(3x) = \cos(2x)$ entirely in terms of $\sin(x).$ Substitute $z = \sin(x)$ and simplify to obtain a cubic equation in $z.$

You should be able to spot a solution to this cubic equation and then use long division to obtain a quadratic which you can solve and then have all three solutions to the cubic equation in $z.$ Decide which of these solutions give you a value of $x$ with $0 < x < 90^o .$


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