SOLUTION: Good evening, I need help to understand step by step this equation: sin(4x + 20°) + sin(2x + 10°) = 0 My first solution is: x = -5° +k.60° and that is correct The second solutio

Algebra ->  Trigonometry-basics -> SOLUTION: Good evening, I need help to understand step by step this equation: sin(4x + 20°) + sin(2x + 10°) = 0 My first solution is: x = -5° +k.60° and that is correct The second solutio      Log On


   

Question 1044552: Good evening, I need help to understand step by step this equation:
sin(4x + 20°) + sin(2x + 10°) = 0
My first solution is: x = -5° +k.60° and that is correct
The second solution should be: x = 85° +k.180° but I cannot reach it
Can somebody explain me how to solve? Thank you very much RB
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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Good evening, I need help to understand step by step this equation:
sin(4x + 20°) + sin(2x + 10°) = 0
My first solution is: x = -5° +k.60° and that is correct
The second solution should be: x = 85° +k.180° but I cannot reach it
Can somebody explain me how to solve? Thank you very much RB
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Notice that  4x + 20° = 2*(2x + 10°).
So you actually have an equation

sin%282%2Aalpha%29+%2B+sin%28alpha%29 = 0

with alpha = 2x + 10°.

Next, as you probably know,  sin%282%2Aalpha%29 = 2*sin(alpha)*cos(alpha), and, therefore, your equation takes the form

2%2Asin%28alpha%29%2Acos%28alpha%29+%2B+sin%28alpha%29 = 0,   or

sin%28alpha%29%2A%282%2Acos%28alpha%29+%2B+1%29 = 0.

It deploys in two independent equations


1.  sin(2x+10°) = 0  --->  2x+10° = 90° or 2x+10° = 180° or 2x+10° = 360° or 2x+10° = 540° --->  

    x = 40°  or  x = 85°  or  x = 175°  or  x = 265°.


2.  cos(2x+10°) = -1%2F2  --->  2x+10° = 120° or 2x+10° = 240° or 2x+10° = 480°  or 2x+10° = 600° --->

    x = 55°  or  x = 115°  or  x = 235°  or  x = 285°.


Answer.  x = 40° or x = 85° or x = 175° or x = 265° or
         x = 55° or x = 115° or x = 235° or x = 285°.