Solving the System of Equations Involving Sine and Cosine: A Step-by-Step Guide
In this guide, we will explore the method to solve a system of equations that involves trigonometric functions, specifically sine and cosine. The system consists of two equations:
Step 1: Express y in Terms of x
Given the first equation:
$$ x y frac{2pi}{3} $$Express (y) in terms of (x):
$$ y frac{2pi}{3} - x $$Step 2: Substitute y into the Second Equation
The second equation is:
$$ frac{sin x}{sin y} 2 $$Substitute (y frac{2pi}{3} - x) into the second equation:
$$ frac{sin x}{sinleft(frac{2pi}{3} - xright)} 2 $$Step 3: Use the Sine Subtraction Formula
Apply the sine subtraction formula:
$$ sinleft(frac{2pi}{3} - xright) sinleft(frac{2pi}{3}right)cos x - cosleft(frac{2pi}{3}right)sin x $$Knowing the values:
$$ sinleft(frac{2pi}{3}right) frac{sqrt{3}}{2}, quad cosleft(frac{2pi}{3}right) -frac{1}{2} $$Substitute these values into the formula:
$$ sinleft(frac{2pi}{3} - xright) frac{sqrt{3}}{2}cos x - left(-frac{1}{2}right)sin x frac{sqrt{3}}{2}cos x frac{1}{2}sin x $$Step 4: Substitute Back into the Equation
Substitute back into the second equation:
$$ frac{sin x}{frac{sqrt{3}}{2}cos x frac{1}{2}sin x} 2 $$Step 5: Cross-Multiply to Eliminate the Fraction
Cross-multiplying gives:
$$ sin x 2left(frac{sqrt{3}}{2}cos x frac{1}{2}sin xright) $$This simplifies to:
$$ sin x sqrt{3}cos x sin x $$Step 6: Rearranging the Equation
Subtract (sin x) from both sides:
$$ 0 sqrt{3}cos x $$This implies:
$$ cos x 0 $$The solutions for (cos x 0) are:
$$ x frac{pi}{2} npi, quad n in mathbb{Z} $$Step 7: Solving for x and y
If ( x frac{pi}{2} ), then:
$$ y frac{2pi}{3} - frac{pi}{2} frac{4pi}{6} - frac{3pi}{6} frac{pi}{6} $$If ( x frac{3pi}{2} ), then:
$$ y frac{2pi}{3} - frac{3pi}{2} frac{4pi}{6} - frac{9pi}{6} -frac{5pi}{6} $$Summary of Solutions
The solutions to the system are:
$( x, y ) left( frac{pi}{2}, frac{pi}{6} right) $( x, y ) left( frac{3pi}{2}, -frac{5pi}{6} right)These values satisfy both equations in the system.