Math  /  Trigonometry

QuestionSolve sin(2x)cos(5x)cos(2x)sin(5x)=0.4\sin (2 x) \cos (5 x)-\cos (2 x) \sin (5 x)=-0.4 for the smallest positive solution. x=x=
Give your answer accurate to two decimal places.

Studdy Solution
Solve for x x by dividing by 3:
For the first solution:
x=0.4115+2kπ3x = \frac{0.4115 + 2k\pi}{3}
For the second solution:
x=π0.4115+2kπ3x = \frac{\pi - 0.4115 + 2k\pi}{3}
Find the smallest positive x x :
For k=0 k = 0 :
x0.411530.1372x \approx \frac{0.4115}{3} \approx 0.1372
For the second solution with k=0 k = 0 :
xπ0.411530.9108x \approx \frac{\pi - 0.4115}{3} \approx 0.9108
The smallest positive solution is:
x0.14x \approx 0.14
The smallest positive solution for x x is:
0.14 \boxed{0.14}

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