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Solve the equation 5sinx=4cosx

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Inna Shpiro eNotes educator | Certified Educator

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To solve the equation 5sinx = 4cosx, first, bring all terms to one side of the equations, so that the other side becomes 0:

5sinx - 4cosx = 0.

Then, factor out 4cosx from the right hand side:

4cosx((5sinx)/(4cosx) - 1) = 0.

The product equals zero when either of the factors is zero, so

cos(x) = 0 or (5sinx)/(4cosx) - 1 = 0.

If cos(x) = 0, then sin(x) also has to be zero in order for the original equation to be true. However, it is impossible for cos(x) and sin(x) to be zero simultaneously (for the same values of x). Therefore, cos(x) = o does not yield valid solutions.

If (5sinx)/(4cosx) - 1 = 0, then

5/4 tan(x) = 1, or

tan(x) = 4/5

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giorgiana1976 | Student

This is an homogenous equation and we'll create the tangent function to solve it

5sinx=4cosx

We'll divide by 5:

sin x = ( 4/5) cos x

Now, we'll create the tangent function by dividing both sides by cos x:

sinx/cosx = 4/5

tan x= 4/5

We've get an elementary equation:

x = arctan (4/5) + k*pi

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