The least positive root of the equation $\cos 3x+\sin 5x=0$ is,
- $\dfrac{3 \pi}{16}$
- $\dfrac{ \pi}{16}$
- $\dfrac{7 \pi}{16}$
- $\dfrac{9 \pi}{16}$
Reveal answer
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A
Correct answer
Explanation
We can rewrite the equation as sin(5x) = -cos(3x) = sin(3x - pi/2). This leads to two sets of general solutions, one of which is 5x = (2k + 1)pi - (3x - pi/2), which simplifies to 8x = (2k + 1)pi + pi/2. Setting k = 0 gives the smallest positive root of x = 3pi/16.
AI explanation
Applying the transformation formula for sine, sin(5x) equals cos(pi/2 - 5x). The equation cos(3x) + sin(5x) = 0 becomes cos(3x) = -cos(pi/2 - 5x), which equals cos(pi + pi/2 - 5x). Equating the angles gives 3x = 3pi/2 - 5x or 3x = -3pi/2 + 5x, leading to the general solutions x = 3pi/16 + n*pi or x = 3pi/4 + n*pi. Testing these equations for integer values of n, the smallest positive value generated is 3pi/16.