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Is the sine function even or odd?\newlineChoices:\newline[A]even\text{[A]even}\newline[B]odd\text{[B]odd}

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Symmetry and periodicity of trigonometric functionsright chevron icon

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Q. Is the sine function even or odd?\newlineChoices:\newline[A]even\text{[A]even}\newline[B]odd\text{[B]odd}
  1. Definition of Even and Odd Functions: To determine if the sine function is even or odd, we need to check if it satisfies the definition of either an even function or an odd function. An even function satisfies the condition f(x)=f(−x)f(x) = f(-x) for all xx in its domain, while an odd function satisfies the condition f(−x)=−f(x)f(-x) = -f(x) for all xx in its domain. We will test the sine function against these conditions.
  2. Testing the Sine Function at a Positive Angle: Let's consider the sine function at a positive angle θ\theta. The sine of θ\theta is given by sin⁡(θ)\sin(\theta).
  3. Testing the Sine Function at a Negative Angle: Now let's consider the sine function at a negative angle −θ-\theta. The sine of −θ-\theta is given by sin⁡(−θ)\sin(-\theta).
  4. Properties of Trigonometric Functions: By the properties of trigonometric functions, we know that sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta). This is because sine is an odd function with respect to the origin on the unit circle, meaning that it reflects across the origin.
  5. Conclusion: Sine Function is Odd: Since sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta) holds true for all values of θ\theta in the domain of the sine function, we can conclude that the sine function satisfies the condition for being an odd function.

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