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Find the number of solutions of the determinant equation | sin(theta) cos(theta) sin(2*theta) | | cos(theta+pi/6) cos(theta+2pi/3) -sin(2*theta+pi/3) | = 0 | -cos(theta-pi/6) cos(theta-2pi/3) sin(2*theta+2pi/3) | in [0, 2*pi].
- 2
- 4
- 8
- infinite
Correct answer: infinite
Solution
The determinant can be shown to equal zero for all values of theta by expanding using trigonometric identities and angle addition formulas — the rows are linearly dependent for all theta. Hence the equation holds for all theta in [0, 2*pi], giving infinitely many solutions.
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