Understanding Cos 2x | The Double-Angle Identity and its Calculation


In mathematics, cos 2x refers to the cosine of twice the angle x

In mathematics, cos 2x refers to the cosine of twice the angle x. To fully understand cos 2x, let’s break it down.

Cosine Function:
The cosine function, often denoted as cos, is a mathematical function that relates an angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. In a unit circle, the cosine of an angle is the x-coordinate of the point on the unit circle that corresponds to that angle.

Double-Angle Identity:
The double-angle identity is a trigonometric identity that relates the cosine of an angle to the cosine of twice that angle. The double-angle identity for the cosine function is given by the formula:

cos 2x = cos^2(x) – sin^2(x)

or alternatively:

cos 2x = 2 cos^2(x) – 1

In both formulas, cos^2(x) refers to the square of the cosine of x, and sin^2(x) refers to the square of the sine of x.

Using the double-angle identity, you can calculate the value of cos 2x by substituting the value of x into the formula. For example, if x = 30 degrees, then 2x = 60 degrees, and you can use the double-angle identity to find cos 60 degrees.

cos 60 degrees = 2 cos^2(30 degrees) – 1

Now, you can calculate cos^2(30 degrees) by finding the cosine of 30 degrees and squaring it. The cosine of 30 degrees is sqrt(3)/2, and when squared, you get:

cos^2(30 degrees) = (sqrt(3)/2)^2 = 3/4

Substituting this value into the formula:

cos 60 degrees = 2 * (3/4) – 1 = 6/4 – 1 = 1/2

Therefore, cos 60 degrees is equal to 1/2.

Similarly, you can compute the value of any cos 2x using the double-angle identity by substituting the value of x and working through the equation.

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