Solving Trigonometric Equations: sin2 θ - cos θ cos2 θ
Introduction
This article will provide a detailed explanation and solution steps for the given trigonometric equation sin^2 θ - cos θ cos^2 θ. Trigonometric equations are fundamental in various mathematical and engineering applications, making it crucial to understand their solution methodologies.
Solution Steps
Let's start with the given equation:
sin^2 θ - cos θ cos^2 θ
First, let's manipulate the equation by bringing all terms to one side:
sin^2 θ - cos^2 θ - cos θ 0
We can further simplify and rearrange the equation as:
2cos^2 θ cos θ - 1 0
This is a quadratic equation in terms of cos θ.
Using Quadratic Formula
The general form of a quadratic equation is ax^2 bx c 0. Here, we have:
a 2, b 1, c -1
The quadratic formula is:
x (-b ± √(b^2 - 4ac)) / 2a
Substitute the values:
cos θ (-1 ± √(1^2 - 4(2)(-1))) / 2(2)
Calculate the discriminant:
Discriminant 1 8 9
Thus:
cos θ (-1 ± √9) / 4
Solving for the roots:
cos θ (-1 3) / 4 1/2 and cos θ (-1 - 3) / 4 -1
Hence, we have two cases for θ:
Cosine Values to Angles
1. cos θ -1
θ arccos(-1) 180° or π
2. cos θ 1/2
θ arccos(1/2) 60° or π/3
Considering the periodicity of sine and cosine functions, we can find all possible solutions within one period [0, 2π]. Let's consider the angles:
θ 180°, 60°, and their corresponding coterminal angles:
Coterminal Angles
Coterminal angles within the range [0, 360°] for θ 60° and θ 180° are:
- For θ 60°: 60°, 300°
- For θ 180°: 180°, 540° (which is coterminal with 180°)
Closure
We have solved the given trigonometric equation, sin^2 θ - cos θ cos^2 θ, and found the angles that satisfy this equation within the domain [0, 2π] and their coterminal angles within [0, 360°]. The derived solutions include:
θ 60°, 300°, 180°
This article has provided a comprehensive method to solve such equations, utilizing the properties of trigonometric functions and quadratic equations. These techniques are essential in various calculative and theoretical applications in mathematics and beyond.