91Ó°ÊÓ

Trigonometric identities are like mathematical shortcuts that help simplify and manipulate trigonometric expressions. They are fundamental tools in mathematics, especially when working with angles and solving equations. In this exercise, we dealt with a specific type of identity called the "double-angle identity." Double-angle identities provide relationships between trigonometric functions of double angles (e.g., \(2x\)) and the functions of the original angles (e.g., \(x\)).

These identities are valuable because they allow us to transform and simplify expressions, such as converting expressions involving \(\sin^2x\) to ones involving \(\cos^2x\), and vice versa.

• For sine, the double-angle identity is: \(\sin(2x) = 2\sin x \cos x\).
• For cosine, there are a few forms, but the one used here is: \(\cos(2x) = 1 - 2\sin^2x\).

In this exercise, we used these identities to rewrite and simplify the given expression. Understanding and applying these identities helps develop better skills in trigonometry.

The sine function is a fundamental trigonometric function used to describe relationships in right-angled triangles and oscillatory phenomena. It is denoted by \(\sin(x)\) and is defined as the ratio of the length of the side opposite the angle \(x\) to the hypotenuse of the triangle.

One of the critical features of the sine function is its periodic behavior, repeating every \(2\pi\) radians or 360 degrees. This quality makes it crucial in fields involving cycles, such as physics and engineering. In trigonometric identities, especially within double-angle formulas, the sine function links angles through its square, \(\sin^2x\).

• For example in double-angle formulas: \(\cos(2x) = 1 - 2\sin^2x\).

In the given exercise, knowing the double-angle identities involving sine allowed us to replace parts of an expression with a form that included cosine, demonstrating the interrelated nature of these functions.

The cosine function is another crucial component in the study of trigonometry. It complements the sine function and provides a way to describe angles and triangles. The cosine of an angle \(x\), written as \(\cos(x)\), is the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.

Cosine, like sine, is also a periodic function, with a period of \(2\pi\) radians. It is crucial when working with angles because of its relationship with sine and its presence in double-angle formulas.

• For instance, one form of the double-angle identity for cosine is: \(\cos(2x) = 1 - 2\sin^2x\).
• Another form is: \(\cos(2x) = 2\cos^2x - 1\).

In the process of solving the exercise, we transformed an expression involving \(\sin^2x\) with the help of the identity to a more manageable form that involves \(\cos^2x\), simplifying the computation. This showcases the utility of these identities in simplifying complex trigonometric expressions.