Question

solve sin(2x)cos(6x)−cos(2x)sin(6x)=−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 \(\sin\theta = k\), then \(\theta=\sin^{-1}(k)+2n\pi\) or \(\theta=\pi-\sin^{-1}(k)+2n\pi\), \(n\in\mathbb{Z}\).
For \(\sin(4x)=0.85\), \(4x=\sin^{-1}(0.85)+2n\pi\) or \(4x=\pi-\sin^{-1}(0.85)+2n\pi\).
\(\sin^{-1}(0.85)\approx1.0297\) (in radians).
First case: \(4x = 1.0297+2n\pi\).
Second case: \(4x=\pi - 1.0297+2n\pi\approx3.1416-1.0297 + 2n\pi=2.1119+2n\pi\).

Step3: Solve for \(x\) (smallest positive solution)

For the first case, when \(n = 0\), \(x=\frac{1.0297}{4}=0.2574\).
For the second case, when \(n = 0\), \(x=\frac{2.1119}{4}=0.5280\).

Answer:

\(x = 0.26\)