Explain why some trigonometric identities have restrictions or non-permissible
values. What are the restrictions for each of the reciprocal trigonometric ratios?
help
If a trig function has a value of zero for a given angles, then clearly its
reciprocal will be undefined.
e.g. cos 90° = 0
then sec 90 = 1/0 which is undefined
e.g. tan 180° = 0, so cot 180° will be undefined.
So, whenever the graph of a trig function crosses the x-axis, that is,
when its value is zero, then its reciprocal function will be undefined for
that value of x
thank you so much!
Some trigonometric identities have restrictions or non-permissible values because certain values of the input variables may lead to undefined or ambiguous results. The restrictions for each of the reciprocal trigonometric ratios are as follows:
1. The reciprocal of sine (cosecant or csc): This ratio is undefined when the sine value is zero, as dividing by zero is not allowed. Therefore, the restriction is that the input angle should not be equal to an integer multiple of π (where sinθ = 0).
2. The reciprocal of cosine (secant or sec): This ratio is undefined when the cosine value is zero. The restriction is that the input angle should not be equal to an odd multiple of π/2 (where cosθ = 0).
3. The reciprocal of tangent (cotangent or cot): This ratio is undefined when the tangent value is zero. The restriction is that the input angle should not be equal to an integer multiple of π (where tanθ = 0).
It is important to note that these restrictions are specific to the reciprocal trigonometric ratios and other trigonometric identities may have different restrictions or non-permissible values.
Trigonometric identities are mathematical equations that establish relationships between various trigonometric functions. Some of these identities have restrictions or non-permissible values because they create undefined or nonsensical results when certain values are used. These restrictions are typically related to the properties and behaviors of the trigonometric functions involved.
Let's discuss the restrictions for each of the reciprocal trigonometric ratios, which are the cosecant, secant, and cotangent functions:
1. Cosecant (csc) function: The cosecant function is the reciprocal of the sine function, represented as csc(x) or 1/sin(x). This function has restrictions when the value of sin(x) equals zero, because dividing by zero is undefined. Therefore, for the cosecant function, non-permissible values occur when x is an integer multiple of π, such as x = 0, π, 2π, etc.
2. Secant (sec) function: The secant function is the reciprocal of the cosine function, denoted as sec(x) or 1/cos(x). Similar to the cosecant function, the secant function has restrictions when cos(x) is equal to zero, resulting in undefined values. Therefore, for the secant function, non-permissible values occur when x is an odd multiple of π/2, such as x = π/2, 3π/2, 5π/2, etc.
3. Cotangent (cot) function: The cotangent function is the reciprocal of the tangent function, written as cot(x) or 1/tan(x). The cotangent function has restrictions when the value of tan(x) equals zero because dividing by zero is undefined. Hence, for the cotangent function, non-permissible values occur when x is a multiple of π, such as x = 0, π, 2π, etc.
It is essential to be aware of these restrictions while working with trigonometric identities to ensure valid and meaningful solutions.