Question 1209808
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°**