Trigonometry | Understanding and Simplifying the Sine Function with a 45-Degree Angle in a Right Triangle

sin 45°

In trigonometry, the sine function (denoted as sin) relates the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right triangle

In trigonometry, the sine function (denoted as sin) relates the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right triangle.

When we have an angle of 45 degrees, we can construct a right triangle where one acute angle is 45 degrees. In this triangle, the side opposite the 45-degree angle is equal to the adjacent side because it is an isosceles triangle. In an isosceles right triangle, the two legs are congruent, which means that the length of the side opposite the angle is equal to the length of the adjacent side.

By convention, we define the hypotenuse of this particular triangle as the square root of 2. Therefore, the ratio of the length of the side opposite the 45-degree angle to the hypotenuse is 1/√2.

To simplify this ratio, we rationalize the denominator by multiplying both the numerator and the denominator by √2:

(sin 45°) = (1/√2) * (√2/√2) = √2/2

So, sin 45° = √2/2.

This value is a commonly used exact value in trigonometry and is equal to approximately 0.707.

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