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Is Sin | x | = | Sin x |?

In this article, we are going to learn if the value of sine of modulus of X is equal to the value of the modulus value or the absolute value of X.

Understanding the Sine Function

The sine function, denoted as sin x, is a periodic function that describes the relationship between the angles of a right triangle and the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the context of a unit circle, sin x represents the y-coordinate of a point on the circle corresponding to an angle x.

The Modulus or Absolute Value Function:

The modulus or absolute value function, denoted as | x |, is a mathematical operation that returns the non-negative value of a real number x. In other words, | x | is the distance of x from the origin on the real number line. For positive values of x, | x | is equal to x, and for negative values of x, | x | is equal to -x.

Analyzing Sin | x |

Now, let’s consider sin | x |. The modulus function ensures that the input is non-negative. Therefore, sin | x | essentially implies the sine of the non-negative value of x. This means that whether x is positive or negative, sin | x | always calculates the sine of the positive value of x.

Analyzing | Sin x |

On the other hand, | sin x | calculates the absolute value of the sine of x. The sine function produces values between -1 and 1 for any real number x. Taking the absolute value ensures that the result is non-negative.

Comparing Sin | x | and | Sin x |

To investigate whether sin | x | is equal to | sin x |, we need to consider the cases where x is positive, negative, or zero.

  1. For x ≥ 0: In this case, sin | x | is equivalent to sin x since the modulus function does not alter positive values. Similarly, | sin x | is also sin x for positive values. Therefore, sin | x | equals | sin x | when x is non-negative.
  2. For x < 0: When x is negative, sin | x | calculates the sine of the positive value of x, while | sin x | calculates the absolute value of the sine of x. Since the sine function is an odd function (sin(-x) = -sin x), sin | x | and | sin x | are equal in magnitude but opposite in sign.

Conclusion

In conclusion, the relationship between sin | x | and | sin x | depends on the sign of x. When x is non-negative, the two expressions are equal. However, when x is negative, sin | x | and | sin x | have the same magnitude but differ in sign. Understanding the properties of the sine function and the modulus function is crucial for comprehending this relationship in the realm of trigonometry.

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