Solve 3sinx 4cosx 2 Exact solutions are not possible for t

Solve 3sinx + 4cosx = 2. Exact solutions are not possible for this problem. To complete the solution, you will need to use a calculator. Use radian mode. The answers are: x = 1.80, 5.76. Show all work leading up to these answers.

Solution

given 3sinx + 4cosx = 2

now taking taking 4cosx to the right hand side

so 3sinx = 2 -4cosx

now squaring on both sides

(3sinx)^2 = (2 - 4cosx)^2

9sin²x = (2²)+ (4cosx)² - 2(2) (4cosx)

9sin²x = 4 + (4cosx)^2 - 2(2) (4cosx) [ formula:(a-b)^2 =a^2 +b^2 -2ab here a =2 and b=4cosx ]

9sin² x = 4 + 16cos² x - 16cosx

Since most terms are cosine, we can substitute 1-cos²x for sin²x to get:
9(1-cos²x) = 4 - 16cosx + 16cos²x
9 - 9cos²x = 4 - 16cosx + 16cos²x

4 - 16cosx + 16cos²x - 9 + 9cos²x =0

25cos²x -16cosx -5=0

Not factorable, so we will use quadratic formula to solve for cosx:

cosx = -(-16)±(256+500) / 50

cosx = 16±(756) / 50

cosx = 16±27.5 / 50

so cosx = (16+27.5 ) / 50 or cosx = (16 -27.5) / 50

cosx = 43.5/50 or cosx = -11.5/50

   cosx= 0.87 or cosx= -0.23

  x = arccos -0.23 = 103.3° or 360° - arccos -0.23 = 256.7°

x = arccos 0.87 = 29.5° or 360° - arccos .87 = 330.5°

But wait, these answers must be checked in the original equation since we did square both sides of the equation, causing possible extraneous solutions.

Check: 103.3° works
256.7° does not work
29.5° does not work
330.5° works
So you have two valid solutions {103,3°, 209.6°}

now we have to convert these degree into radians

that is 180 degree is pi

103 degree is 1.796 = 1.80 (approximatly)

similerly

x = 330.5 * pi /180 = 330.5 *3.14 /180 =5.7653

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