Domains of Trigonometric functions and of Fundamental Identities

 The domains of trigonometric functions and fundamental identities depend on the specific function or identity being considered. Here's a brief overview:

  1. Domains of Trigonometric Functions:

    • Sine Function (sin(x)): The domain of the sine function is all real numbers. In other words, you can input any real number into the sine function and get a valid output.
    • Cosine Function (cos(x)): Similar to the sine function, the domain of the cosine function is also all real numbers.
    • Tangent Function (tan(x)): The tangent function is undefined at odd multiples of π/2 (e.g., π/2, 3π/2, -π/2, etc.) because at these points, cosine is zero, making the fraction undefined. Therefore, the domain of the tangent function is all real numbers excluding odd multiples of π/2.
    • Cosecant Function (csc(x)): The cosecant function is the reciprocal of the sine function, so it is undefined wherever sine is zero. Therefore, the domain of the cosecant function is all real numbers excluding integer multiples of π.
    • Secant Function (sec(x)): Similar to the cosecant function, the secant function is the reciprocal of the cosine function, so it is undefined wherever cosine is zero. Therefore, the domain of the secant function is all real numbers excluding odd multiples of π/2.
    • Cotangent Function (cot(x)): The cotangent function is the reciprocal of the tangent function, so it is undefined wherever tangent is zero. Therefore, the domain of the cotangent function is all real numbers excluding integer multiples of π.

  2. Fundamental Trigonometric Identities:

    • Fundamental trigonometric identities are equations involving trigonometric functions that are true for all values in their respective domains.
    • Examples of fundamental trigonometric identities include:
      • Pythagorean identities: sin⁡2(�)+cos⁡2(�)=1
      • Reciprocal identities: csc⁡(�)=1sin⁡(�), sec⁡(�)=1cos⁡(�), cot⁡(�)=1tan⁡(�)
      • Quotient identities: tan⁡(�)=sin⁡(�)cos⁡(�), cot⁡(�)=cos⁡(�)sin⁡(�)

These identities are valid for all values in their respective domains. It's important to note that these are just a few examples, and there are several other trigonometric identities that are used in various contexts.

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