The Sine Function: Definition, Graph, Values And Identities – A Comprehensive Guide.

sinx

cosx

sinx is a trigonometric function that represents the ratio of the length of the side opposite to a given angle (x) in a right triangle to the length of the hypotenuse. It is defined for all real values of x and takes values between -1 and 1.

The graph of sinx is periodic with a period of 2π. This means that the value of sinx repeats itself after every 2π radians or 360 degrees. The function is also symmetric about the origin, which means that sin(-x) = -sinx.

Some important values of sinx are:

– sin(0) = 0
– sin(π/6) = 1/2
– sin(π/4) = √2/2
– sin(π/3) = √3/2
– sin(π/2) = 1
– sin(π) = 0
– sin(3π/2) = -1

The trigonometric identities associated with sinx are:

– sin(-x) = -sinx (symmetry)
– sin(x ± y) = sinxcosy ± cosxsiny (sum and difference)
– sin2x = 2sinxcosx (double angle)
– sin(x/2) = ± √((1-cosx)/2) (half angle)

The sine function has several applications in mathematics, including in geometry, calculus, and differential equations. It is also used in physics to describe oscillatory motion, such as the motion of a simple pendulum or a vibrating string.

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