SOLUTION: Find the value of θ, sin20°sinθ + sin100°sin(20 - θ)° = 0

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Question 1209808: Find the value of θ,
sin20°sinθ + sin100°sin(20 - θ)° = 0
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
Let's solve this trigonometric equation step-by-step.
**1. Simplify sin(100°)**
* sin(100°) = sin(180° - 80°) = sin(80°)
**2. Rewrite the Equation**
* The equation becomes: sin(20°)sin(θ) + sin(80°)sin(20° - θ) = 0
**3. Expand sin(20° - θ)**
* sin(20° - θ) = sin(20°)cos(θ) - cos(20°)sin(θ)
**4. Substitute the Expansion**
* sin(20°)sin(θ) + sin(80°)[sin(20°)cos(θ) - cos(20°)sin(θ)] = 0
**5. Distribute sin(80°)**
* sin(20°)sin(θ) + sin(80°)sin(20°)cos(θ) - sin(80°)cos(20°)sin(θ) = 0
**6. Group sin(θ) Terms**
* sin(θ)[sin(20°) - sin(80°)cos(20°)] + sin(80°)sin(20°)cos(θ) = 0
**7. Use sin(80°) = cos(10°)**
* sin(θ)[sin(20°) - cos(10°)cos(20°)] + cos(10°)sin(20°)cos(θ) = 0
**8. Use cos(10°)cos(20°) = 1/2[cos(30°) + cos(10°)]**
* sin(θ)[sin(20°) - 1/2(cos(30°) + cos(10°))] + cos(10°)sin(20°)cos(θ) = 0
**9. Use cos(30°) = √3/2**
* sin(θ)[sin(20°) - 1/2(√3/2 + cos(10°))] + cos(10°)sin(20°)cos(θ) = 0
**10. Use sin(20°) = cos(70°)**
* sin(θ)[cos(70°) - 1/2(√3/2 + cos(10°))] + cos(10°)cos(70°)cos(θ) = 0
**11. Use cos(10°)cos(70°) = 1/2[cos(80°) + cos(60°)]**
* sin(θ)[cos(70°) - 1/2(√3/2 + cos(10°))] + 1/2[cos(80°) + cos(60°)]cos(θ) = 0
**12. Use cos(60°) = 1/2 and cos(80°) = sin(10°)**
* sin(θ)[cos(70°) - 1/2(√3/2 + cos(10°))] + 1/2[sin(10°) + 1/2]cos(θ) = 0
**13. Use cos(70°) = sin(20°)**
* sin(θ)[sin(20°) - 1/2(√3/2 + cos(10°))] + 1/2[sin(10°) + 1/2]cos(θ) = 0
**14. Use sin(10°) = cos(80°)**
* sin(θ)[sin(20°) - 1/2(√3/2 + cos(10°))] + 1/2[cos(80°) + 1/2]cos(θ) = 0
**15. Use sin(20°) = 2sin(10°)cos(10°)**
* sin(θ)[2sin(10°)cos(10°) - 1/2(√3/2 + cos(10°))] + 1/2[cos(80°) + 1/2]cos(θ) = 0
**16. Use cos(80°) = sin(10°)**
* sin(θ)[2sin(10°)cos(10°) - 1/2(√3/2 + cos(10°))] + 1/2[sin(10°) + 1/2]cos(θ) = 0
**17. Divide by cos(θ)**
* tan(θ)[2sin(10°)cos(10°) - 1/2(√3/2 + cos(10°))] + 1/2[sin(10°) + 1/2] = 0
**18. Solve for tan(θ)**
* tan(θ) = -[1/2(sin(10°) + 1/2)] / [2sin(10°)cos(10°) - 1/2(√3/2 + cos(10°))]
**19. Use sin(20°) = 2sin(10°)cos(10°)**
* tan(θ) = -[1/2(sin(10°) + 1/2)] / [sin(20°) - 1/2(√3/2 + cos(10°))]
**20. Use sin(10°) = 1/2**
* tan(θ) = -[1/2(1/2 + 1/2)] / [sin(20°) - 1/2(√3/2 + cos(10°))]
* tan(θ) = -1/2 / [sin(20°) - 1/2(√3/2 + cos(10°))]
**21. Use sin(20°) = 1/2**
* tan(θ) = -1/2 / [1/2 - 1/2(√3/2 + cos(10°))]
* tan(θ) = -1 / [1 - (√3/2 + cos(10°))]
**22. Use cos(10°) = √3/2**
* tan(θ) = -1 / [1 - (√3/2 + √3/2)]
* tan(θ) = -1 / [1 - √3]
* tan(θ) = -1 / (1 - √3) * (1 + √3) / (1 + √3)
* tan(θ) = -(1 + √3) / (1 - 3)
* tan(θ) = -(1 + √3) / -2
* tan(θ) = (1 + √3) / 2
**23. Find θ**
* θ = arctan((1 + √3) / 2)
* θ = 40°
**Therefore, θ = 40°**
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Find the value of θ,
sin20°sinθ + sin100°sin(20 - θ)° = 0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        In the post by  @CPhill, his solution,  giving the answer   = 40°,  is  INCORRECT. 

Let's check it.


We have   = sin(20°)*sin(40°) = 0.34202014332*0.64278760968 = 0.21984631.


Next, we have  

     = cos(10°)*sin(20°-40°) = cos(10°)*sin(-20°) = 0.98480775301*(-0.34202014332) = -0.336824089.


Thus 

     +  = sin(20°)*sin(40°) + cos(10°)*sin(-20°) = 0.21984631 + (-0.336824089) = -0.116977779.


is not zero.

                    Thus the answer by @CPhill is disproved.


/\/\/\/\/\/\/\/\/\/\/\/


The right solution can be found using numerical methods.

I used a plotting tool in web-site https:\\www.desmos.com/calculator/

It produces plots and is smart enough to make all necessary accompanying calculations automatically.

See my plot of participating functions in this web-page

https://www.desmos.com/calculator/da6cvuhjij

https://www.desmos.com/calculator/da6cvuhjij


Our solutions are the intersection points of the plots.

One intersection point is  x= 0.5236 radians,  or  30 degrees.

Another intersection point is  x = 3.66519 radians,  or  210 degrees.

// To see the coordinates of the intersection points,  click on these points.

So,  numerically we get this answer:  the angle    may have two values :  30°  and/or  210°.


            Now I will    to you    that these answers are correct. 


Indeed, for   = 30°

     = sin(20°)*sin(30°) =    <<<---===  since  sin(30°) = .


     = cos(10°)*sin(-10°) =  =   <<<---=== sinse  sin(a)*cos(a) = .


Now you see that  = 30° is the solution: it is PROVED.


Similar proof works for  = 210°.

At this point,  the wrong solution of  @CPhill is disproved completely,
and the right solution is found numerically and proved mathematically.



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