Fundamentals of Engineering Review: Trigonometry_Prob 1

HOW TO SOLVE SIMPLE TRIGONOMETRIC QUADRATIC EQUATIONS BY FACTORING

Trigonometric quadratic equations, such as those involving sin(x) or cos(x), can often be solved efficiently by factoring. Here’s a step-by-step guide on how to solve these equations using the example:

sin²(x) — sin(x) = 0

. Step 1: Recognize the Equation Form

The equation sin²(x) — sin(x) = 0 is a quadratic form where sin(x) is treated as the variable. Recognize that it resembles a standard quadratic equation of the form:

ax² + bx + c = 0.

However, instead of x, we have sin(x).

. Step 2: Factor the Equation

The key to solving such equations is factoring. Start by factoring out the common term:

sin²(x) — sin(x) = sin(x) · (sin(x) — 1).

Factoring breaks the equation into two simpler expressions that multiply to zero. According to the Zero-Product Property, if the product of two factors is zero, then at least one of the factors must be zero.

. Step 3: Set Each Factor to Zero

Next, solve for x by setting each factor equal to zero:

1. First Factor: sin(x) = 0

• To solve sin(x) = 0, we find the values of x where the sine function equals zero within the interval 0 ≤ x < 2π.

• The solutions are:

x = 0, π, 2π.

2. Second Factor: sin(x) — 1 = 0

• Solving sin(x) — 1 = 0 leads to:

sin(x) = 1.

• The value of x where sin(x) = 1 within the interval is:

x = π/2.

. Step 4: Combine All Solutions

Finally, combine all the values obtained from each factor to get the complete set of solutions. In this case, the solutions are:

x = 0, π/2, π, 2π.

. Summary of Steps

• Identify the quadratic form: Recognize that the trigonometric equation resembles a standard quadratic form.

• Factor the equation: Look for common terms and factorize the expression.

• Apply the Zero-Product Property: Set each factor equal to zero and solve separately.

• Solve within the interval: Find all relevant solutions for interval

0 ≤ x < 2π.

• Combine solutions: List all distinct values of x that satisfy the equation.