91Ó°ÊÓ

The Double Angle Formula is a crucial tool in trigonometry. It allows us to express trigonometric functions of twice an angle in terms of the functions of the angle itself. This is particularly useful when verifying identities or simplifying expressions. The formula for cosine is:

• \( \cos 2x = \cos^2 x - \sin^2 x \)
• Alternate forms include \( \cos 2x = 2\cos^2 x - 1 \) and \( \cos 2x = 1 - 2\sin^2 x \)

These variations stem from the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \), which allows us to rewrite \( \cos^2 x \) and \( \sin^2 x \) in terms of each other.
In our exercise, we specifically used the version \( \cos 2x = 1 - 2 \sin^2 x \), showing that \( \cos 6x = 1 - 2 \sin^2 3x \). This manipulation is critical because it directly links the cosine of a double angle to its sine form.

The Cosine Function, denoted as \( \cos \), is one of the foundational trigonometric functions. It arises in various branches of mathematics and physics, especially concerning periodic phenomena like waves and oscillations.
The cosine of an angle in a right triangle is defined as the length of the adjacent side over the hypotenuse. Mathematically, if we consider an angle \( \theta \), we say \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
Cosine values range from -1 to 1 and it shares a periodic nature, repeating its values every \( 2\pi \) radians or 360 degrees. This periodicity enables transformations like those using the double angle formula, which relate cosine to sine in helpful ways.
In the context of our exercise, knowing the properties and transformations of cosine helps in rewriting and verifying trigonometric identities.

The Sine Function, represented by \( \sin \), is another primary trigonometric function. It often pairs with the cosine function to describe wave patterns and circular motion.
In a right triangle, sine is defined as the ratio of the length of the opposite side to the hypotenuse: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). Just like cosine, sine values also oscillate between -1 and 1 with a periodic cycle of \( 2\pi \) radians.
A key identity involving sine function is \( \sin^2 x + \cos^2 x = 1 \). This identity is instrumental for conversions and simplifications involving sine and cosine.
For verification tasks like our original problem, expressing cosine in terms of sine (or vice versa) is often needed. In this case, utilizing \( 2 \sin^2 x = 1 - \cos 2x \) provides a straightforward path to demonstrate the identity \( \cos 6x = 1 - 2 \sin^2 3x \). This highlights the importance of understanding sine in its mathematical context to solve trigonometric equations efficiently.