Learn how to calculate the sine of 45° using right-angled triangle – Step by step guide

sin 45°

√2/ 2

The value of sine 45° is 0.70710678118.

To understand how to calculate the sine of an angle, we need to consider the right-angled triangle. In a right-angled triangle, 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 (the longest side of the triangle).

In the case of 45°, we can draw a right-angled triangle where one angle measures 45°. We know that in such a triangle, the two acute angles (the angles opposite the two shorter sides) are congruent and measure 45° each.

Since we are given the angle measure of 45° and not any of the side lengths of the triangle, we can assume the sides to be of equal length. Let’s label the length of each of the two shorter sides as x.

Now, using the Pythagorean theorem, we can find the length of the hypotenuse of the triangle:

hypotenuse² = x² + x² (as both the shorter sides are of equal length)
hypotenuse² = 2x²
hypotenuse = √(2x²) = x√2

Now, the sine of the 45° angle will be:

sine 45° = (opposite side) / (hypotenuse)
sine 45° = x / (x√2)
sine 45° = (1/√2)
sine 45° = 0.70710678118 (rounded to 11 decimal places)

Therefore, the value of sine 45° is 0.70710678118.

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