How to Find csc 45° | Understanding the Mathematics Behind the Cosecant Function

csc 45°

The abbreviation “csc” stands for cosecant in mathematics

The abbreviation “csc” stands for cosecant in mathematics. The cosecant is a trigonometric function that is the reciprocal of the sine function.

To find csc 45°, we first need to determine the value of the sine for 45°.

The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. In a 45-45-90 degree triangle, the two sides opposite the 45 degree angles are congruent. Hence, if we consider a right triangle with an angle of 45 degrees, the length of the opposite side will be equal to the length of the adjacent side.

Using the Pythagorean theorem, we can determine the lengths of the sides in this triangle. Let’s assume the length of the hypotenuse is 1 since the hypotenuse is the longest side in a right triangle, and we can always scale the sides later.

According to the Pythagorean theorem, we have:
(opposite side)^2 + (adjacent side)^2 = (hypotenuse)^2

Let the length of the opposite and adjacent sides be x. Then we can write:
x^2 + x^2 = 1^2
2x^2 = 1
x^2 = 1/2
x = √(1/2)
x = 1/√2
x = √2/2

Therefore, the sine of 45° is √2/2.

Now, since the cosecant function is the reciprocal of the sine function, we can find csc 45° by taking the reciprocal of √2/2.

Reciprocal of a number x is 1/x, so:
csc 45° = 1 / (√2/2)

To simplify this expression, we multiply the numerator and denominator by 2 (to get rid of the fraction in the denominator):
csc 45° = 2 / √2

We can rationalize the denominator by multiplying the numerator and denominator by √2:
csc 45° = (2 * √2) / (√2 * √2)
csc 45° = (2 * √2) / 2

Finally, we cancel out the common factor of 2:
csc 45° = √2

So, csc 45° is equal to √2.

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