Understanding the Sine of 45 Degrees | Calculation and Application in Trigonometry

sin45º

The sine of 45 degrees, written as sin(45º), represents the ratio of the length of the side opposite the 45-degree angle to the length of the hypotenuse in a right triangle

The sine of 45 degrees, written as sin(45º), represents the ratio of the length of the side opposite the 45-degree angle to the length of the hypotenuse in a right triangle.

To find the sine of 45 degrees, we can use the special right triangle with angles 45 degrees, 45 degrees, and 90 degrees, which is commonly referred to as the 45-45-90 triangle.

In a 45-45-90 triangle, the two legs (sides adjacent to the 45-degree angles) are congruent, and the length of each leg is represented by x. The hypotenuse (side opposite the right angle) has a length of x√2 (the square root of 2 times the length of a leg).

Since the length of the side opposite the 45-degree angle is the same as the length of a leg in a 45-45-90 triangle, we can say that it is also x in our case.

Therefore, the sine of 45 degrees is given by the ratio of the length of the side opposite the angle (x) to the length of the hypotenuse (x√2):

sin(45º) = x / (x√2)

To simplify this expression, we can divide both the numerator and the denominator by x:

sin(45º) = 1 / √2

To rationalize the denominator (eliminate the square root in the denominator), we can multiply both the numerator and the denominator by √2:

sin(45º) = (1 * √2) / (√2 * √2)

This simplifies to:

sin(45º) = √2 / 2

Therefore, the sine of 45 degrees is √2 / 2.

Note: This value is an exact value and is commonly used in trigonometric calculations.

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