The expression sin 2x refers to the sine of double the angle x. It can be understood using the double angle identity in trigonometry, which states that:
sin 2x = 2 sin x cos x
This identity is derived from the sum of angles formula for sine, which is:
sin(a + b) = sin a cos b + cos a sin b
By taking a = x and b = x, it simplifies to:
sin(x + x) = sin x cos x + cos x sin x = 2 sin x cos x
This means that to calculate sin 2x, you multiply the sine of the angle x by the cosine of the same angle x, and then double the result.
For example, if x = 30° (or π/6 radians), then:
- sin 30° = 1/2
- cos 30° = √3/2
Now, applying the double angle identity:
sin 2(30°) = 2 sin 30° cos 30°
sin 60° = 2 × (1/2) × (√3/2) = √3/2
So, sin 2x effectively helps us find the sine of double an angle, which is useful in various applications in mathematics, physics, and engineering.