Understanding the Reciprocal Identity of Cosecant and Its Application in Trigonometry

Reciprocal Identitycsc θ=

The reciprocal identity for the trigonometric function csc (cosecant) is defined as:

csc θ = 1/sin θ

Here, θ represents an angle in a right triangle or on the unit circle

The reciprocal identity for the trigonometric function csc (cosecant) is defined as:

csc θ = 1/sin θ

Here, θ represents an angle in a right triangle or on the unit circle. The csc function is the reciprocal of the sin function, meaning that it is the ratio of the hypotenuse to the length of the side opposite the angle θ in a right triangle, or the y-coordinate on the unit circle.

In terms of the sin function, the reciprocal identity states that the cosecant of an angle is equal to one divided by the sine of that same angle.

For example, let’s consider a right triangle where the angle θ is opposite the side of length 3 and adjacent to the side of length 4. We can use the Pythagorean theorem to find the length of the hypotenuse, which is √(3^2 + 4^2) = 5.

In this case, the sin θ is equal to opposite/hypotenuse = 3/5. Therefore, the reciprocal identity for csc θ states that csc θ = 1/sin θ = 1/(3/5) = 5/3.

So, the csc θ in this example is 5/3, indicating that for this angle, the cosecant value is 5/3, or in other words, the ratio of the length of the hypotenuse to the length of the side opposite the angle θ is 5/3.

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