Understanding cos2x | Exploring the Cosine Function and Double Angle Formula

cos2x

The expression cos2x refers to the cosine of the angle 2x, where x represents any real number

The expression cos2x refers to the cosine of the angle 2x, where x represents any real number. To understand cos2x, it is important to be familiar with the concept of the cosine function.

The cosine function, denoted as cosθ, is a trigonometric function that relates the angle θ to the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is defined for all real numbers θ.

Now, let’s focus on cos2x specifically. We can rewrite cos2x as cos(2x), which means we are taking the cosine of the angle 2x.

The angle 2x can be visualized as a unit circle, where x can be any real number. As x increases or decreases, the angle 2x in the unit circle will change accordingly.

To evaluate cos(2x), we need to use a trigonometric identity called the double-angle formula for cosine. The double-angle formula states:

cos(2θ) = cos^2(θ) – sin^2(θ)

Applying the double-angle formula to cos(2x), we have:

cos(2x) = cos^2(x) – sin^2(x)

This formula allows us to express cos(2x) in terms of sin(x) and cos(x). However, we can simplify this expression further using another trigonometric identity called the Pythagorean identity:

sin^2(x) + cos^2(x) = 1

Rearranging this identity gives us:

sin^2(x) = 1 – cos^2(x)

Substituting this result back into the double-angle formula, we get:

cos(2x) = cos^2(x) – (1 – cos^2(x))

Simplifying further, we have:

cos(2x) = 2cos^2(x) – 1

This is the final simplified expression for cos(2x). It shows that cos(2x) can be expressed solely in terms of cos(x).

In summary, cos2x, or cos(2x), represents the cosine of the angle 2x. It can be computed using the double-angle formula for cosine, which states that cos(2θ) equals 2cos^2(θ) – 1.

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