91Ó°ÊÓ

The Pythagorean identity in trigonometry is a fundamental principle that relates the square of the sine function to the square of the cosine function. It states that for any angle \( x \):

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is particularly useful in simplifying trigonometric equations, as it can be used to substitute one trigonometric function in terms of another.
In the example problem, we see the identity used to replace \( \sin^2 x \) with \( 1 - \cos^2 x \). By doing this, the equation is transformed entirely in terms of \( \cos x \), making it easier to handle.
Being familiar with the Pythagorean identity can significantly simplify many trigonometric problems by reducing them to more manageable forms using substitution.

Quadratic equations are polynomial equations of the second degree, characterized by their standard form:

• \( ax^2 + bx + c = 0 \)

where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable.
The quintessential feature of quadratic equations is the \( x^2 \) term. In our trigonometric problem, after applying the Pythagorean identity, the leftover expressions can be formulated into a quadratic equation involving \( \cos x \).
We transformed \( -\cos^2 x + \cos x = 0 \) into \( \cos^2 x - \cos x = 0 \) by manipulating the terms. This step verifies that quadratic relationships also fit into trigonometric contexts, not just algebraic times.

Factorization is the process of breaking down an equation into simpler components called factors, which when multiplied together give the original equation. In algebra, this is often used to solve polynomial equations.
For our trigonometric quadratic equation \( \cos^2 x - \cos x = 0 \), factorization allows us to express it as a product of its simpler parts:

• \( \cos x(\cos x - 1) = 0 \)

The factored equation indicates that the solutions occur where either \( \cos x = 0 \) or \( \cos x - 1 = 0 \).
This approach reduces the original quadratic problem into linear equations, making it straightforward to solve for \( x \).
Knowing how to factorize is valuable because it reveals the roots of the equation – the key values that satisfy the equation exactly.