Find all values of x in the interval 0 ≤ x ≤ 2pi for 2sin(x)tan(x)=3
This questions involves the use of trigonometric identities, specifically [1] tan(x) = sin(x)/cos(x) and [2] sin2(x) + cos2(x) = 1.
Start by rearranging the given equations to tan(x) = 3/2sin(x) by diving through by 2sin(x). Next rewrite tan(x) using identity [1] above to give sin(x)/cos(x) = 3/2sin(x).Now cross multiply to give 2sin2(x)=3cos(x), this equation needs to all be in terms of cos(x) before it can be solved so identity [2] is needed to do this. Substitute sin2(x) for 1 - cos2(x) which gives 2(1 - cos2(x)) = 3cos(x). Multiply this out and rearrange to give an equation of a quadratic form; 2cos2(x) + 3cos(x) - 2 = 0. Factorise this quadratic to give (2cos(x) - 1)(cos(x) + 2) = 0, to solve this rearrange each bracket to give cos(x) = 1/2 and cos(x) = 2.
To make sure every solution to the equation is found, drawing a rough cos graph can be very helpful, through this you can see that cos(x) cannot equal 2 and therefore rules out this answer. You can also see that cos(x) = 1/2 has two solutions; where x = pi/3 and x = 5pi/3
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