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Problem 19 – Solution

Posted in Uncategorized by Lia on 07/24/2011

The problem:

Solve the equation:

sin{3x} - cos{2x} - sin{x} =0

 

Admission exam to Mathematics,

University of Bucharest, Romania, 19xx

 

Our Solution:

Let’s start by calculating  sin{3x} - sin{x} using formula sin{alpha} -  sin{beta}  = 2 sin({{alpha} - {beta}}/2)cos({{alpha} + {beta}}/2)

Where:  {alpha} = 3x and {beta} = x.

[1]   sin{3x} - sin{x} = 2 sin({{3x} - {x}}/2)cos({{3x} + {x}}/2) = 2 sin({2x}/2)cos({4x}/2) = 2 sin{x}cos{2x}

By replacing [1] in our equation we have:

[2]  2 sin{x}cos{2x} - cos{2x} = 0  {{}{right}{}}  (2 sin{x} -1)cos{2x} =0

From [2], we can see that the expression on the left side of the equation is {{}{0}{}} when:

[3]   2 sin{x} -1 =0 or

[4]   cos{2x} =0

a)  Solving [3] we have: 2 sin{x} = 1  {{}{right}{}}   sin{x} = {1/2}

We know that  sin{x} = {1/2} when

[5]  {x_1} = 30^{circ} = {{pi}/6}  in the first quadrant of the trigonometric circle.

[6]   {x_2} = 150^{circ} = {pi} - {{pi}/6} = {{5{pi}}/6}   in the second quadrant of the trigonometric circle.

{{}{sin{x}}{}} has a positive value therefore {{}{x}{}} can only be in first and second quadrant.

If we consider angles larger than {{}{2{pi}}{}} ( 360^{circ} ), the general results for {{}{x_1}{}} and  {{}{x_2}{}} are:

[7]   tabular{11}{11}{{{x_1} = 2n{pi} + {{pi}/6}}} and  tabular{11}{11}{{{x_2} = 2n{pi} + {{5{pi}}/6}}} where {{}{n}{}} is an integer.

b)  Solving [4] we havecos{2x} = 0  when:

[8]    2x = {{pi}/2} or {x_{3-1}} = {{pi}/4}  in the first quadrant of the trigonometric circle. ( {{pi}/4} = 45^{circ} )

and

[9]   2x = {{3{pi}}/2}  or  {x_{3-2}} = {{3{pi}}/4}  in the third quadrant of the trigonometric circle.  ({{3pi}/4} = 3*{45^{circ}} = 135^{circ} )

If we consider angles larger than {{}{2{pi}}{}} (360^{circ}) and combine the two results, (for {{}{x_{3-1}}{}} and {{}{x_{3-2}}{}} from [8] and [9]), the general result for {{}{x_3}{}}is:

[10]   tabular{11}{11}{{{x_3} = {{(2n +1){pi}}/4}}} where  {{}{n}{}} is an integer and (2n +1) is the general form in which an odd number is shown.

The solutions for this equation are shown in [7] and [10].