Exploring the Double Angle Identity for Sine: Simplifying and Evaluating Trigonometric Expressions

sin 2x

The expression “sin 2x” represents the sine of twice the angle x

The expression “sin 2x” represents the sine of twice the angle x. In trigonometry, there are several important identities that can be used to simplify or evaluate trigonometric expressions.

One of the most useful identities in this case is the double angle identity for sine, which states:

sin 2x = 2sin x cos x

This identity allows us to express sin 2x in terms of sin x and cos x. To understand why this identity holds, we can consider the unit circle.

When we represent an angle x on the unit circle, the coordinates of the corresponding point are (cos x, sin x). If we take twice that angle, we end up at a point with coordinates (cos 2x, sin 2x). The double angle identity for sine tells us that sin 2x can be expressed as twice the product of the sine and cosine of x.

So, given the expression sin 2x, we can rewrite it as 2sin x cos x.

It’s worth noting that this identity can be further manipulated using other trigonometric identities. For example, we can express sin 2x in terms of only sine using the Pythagorean identity:

sin^2 x + cos^2 x = 1

Rearranging this identity, we have:

sin^2 x = 1 – cos^2 x

Plugging this into the double angle identity for sine, we get:

sin 2x = 2sin x cos x
= 2(sqrt(1 – cos^2 x)) cos x

This form can be useful when evaluating trigonometric expressions involving sin 2x.

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