To find the cosecant (csc) of an angle, you need to know that the cosecant is the reciprocal of the sine function. In other words, csc(x) = 1/sin(x).
Here's how to find cosecant, depending on what information you have:
1. If you know the sine of the angle:
- Simply take the reciprocal of the sine value. For example, if sin(x) = 0.5, then csc(x) = 1/0.5 = 2.
2. If you have a right triangle:
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Recall that sine is defined as the opposite side divided by the hypotenuse (SOH).
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Therefore, cosecant is the hypotenuse divided by the opposite side (H/O).
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Formula: csc(θ) = Hypotenuse / Opposite
- Identify the angle θ.
- Identify the hypotenuse (the side opposite the right angle).
- Identify the opposite side (the side opposite to angle θ).
- Divide the length of the hypotenuse by the length of the opposite side.
Example:
Imagine a right triangle where the angle θ is such that the opposite side has a length of 3 and the hypotenuse has a length of 5.
- csc(θ) = 5/3 ≈ 1.67
3. If you know the angle in radians or degrees:
- Use a calculator that has trigonometric functions.
- Find the sine of the angle (sin(x)).
- Calculate the reciprocal of the sine value (1/sin(x)). This gives you the cosecant. Many calculators do not have a dedicated cosecant function, making this the standard method.
4. Understanding the Relationship to Sine:
- Cosecant is undefined when sine is equal to zero. This occurs at integer multiples of π (e.g., 0, π, 2π, etc.). At these points, the cosecant function has vertical asymptotes.
- The range of the cosecant function is (-∞, -1] ∪ [1, ∞). This is because the sine function has a range of [-1, 1], and taking the reciprocal inverts these bounds.
In Summary: The easiest method to find the cosecant is to calculate the sine of the angle and then take the reciprocal of that value (1/sin(x)). In a right triangle, it is the length of the hypotenuse divided by the length of the opposite side.
