Question

solve \\( \sin ( 2 x ) \cos ( 6 x ) - \cos ( 2 x ) \sin ( 6 x ) = - 0.85 \\) for the smallest positive solution.

\\( x = \\)

give your answer accurate to two decimal places.

question help: video

Explanation:

Step1: Use the sine subtraction formula

Recall the formula \(\sin(A - B)=\sin A\cos B-\cos A\sin B\).
Here \(A = 2x\) and \(B=6x\), so \(\sin(2x)\cos(6x)-\cos(2x)\sin(6x)=\sin(2x - 6x)=\sin(-4x)\).
Since \(\sin(-\alpha)=-\sin\alpha\), then \(\sin(-4x)=-\sin(4x)\).
The equation becomes \(-\sin(4x)=- 0.85\), which simplifies to \(\sin(4x)=0.85\).

Step2: Solve for \(4x\)

We know that if \(\sin\theta = a\), then \(\theta=\sin^{-1}(a)+2k\pi\) or \(\theta=\pi-\sin^{-1}(a)+2k\pi\), \(k\in\mathbb{Z}\).
For \(\sin(4x)=0.85\), \(4x=\sin^{-1}(0.85)+2k\pi\) or \(4x=\pi-\sin^{-1}(0.85)+2k\pi\).
First, find \(\sin^{-1}(0.85)\approx1.01\) (in radians).
For \(k = 0\):

• Case 1: \(4x=\sin^{-1}(0.85)\approx1.01\), then \(x=\frac{1.01}{4}=0.2525\).
• Case 2: \(4x=\pi - 1.01\approx3.14 - 1.01=2.13\), then \(x=\frac{2.13}{4}=0.5325\).

Answer:

\(x = 0.25\)