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What is the Exact Value of csc 270?
Introduction
In trigonometry, the cosecant function (csc) is the reciprocal of the sine function. It is defined as the ratio of the hypotenuse to the side opposite the angle in a right triangle. In this article, we will explore the exact value of csc 270 and understand the concept of cosecant in more detail.
Understanding Cosecant
The cosecant function is defined as:
csc θ = 1/sin θ
where θ is the angle in radians.
- Domain: The domain of the cosecant function is all real numbers except for multiples of π/2, where sin θ = 0.
- Range: The range of the cosecant function is all real numbers except for 0.
- Period: The period of the cosecant function is 2π.
Exact Value of csc 270
To find the exact value of csc 270, we need to determine the sine of 270 degrees.
sin 270° = -1
Using the definition of cosecant, we have:
csc 270° = 1/sin 270° = 1/-1 = -1
Interpretation
Therefore, the exact value of csc 270 is -1. This means that in a right triangle with an angle of 270 degrees, the ratio of the hypotenuse to the side opposite the angle is -1. This is because the opposite side is negative due to the angle being in the third quadrant.
Conclusion
In this article, we explored the concept of cosecant and determined the exact value of csc 270 to be -1. Understanding the cosecant function is crucial for solving trigonometry problems and analyzing periodic functions.
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