Understanding the Sine of an Angle | Explaining the Trigonometric Function and its Calculation in a 60-Degree Triangle

sin60

The sine of an angle is a trigonometric function that relates the ratio of the length of the side opposite the angle to the length of the hypotenuse of a right triangle

The sine of an angle is a trigonometric function that relates the ratio of the length of the side opposite the angle to the length of the hypotenuse of a right triangle. In this case, to find sin 60, we need to consider a right triangle where one of the angles is 60 degrees.

To do this, let’s draw a right triangle with one angle measuring 60 degrees. We can label the sides of the triangle as follows:

– The side opposite the 60-degree angle is called the “opposite” side (let’s call it “o”).
– The side adjacent to the 60-degree angle (and opposite the right angle) is called the “adjacent” side (let’s call it “a”).
– The hypotenuse is the longest side of the triangle and we’ll label it as “h”.

Now, in a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. So, we have:

sin(60) = o/h

In trigonometry, we know that for a 30-60-90 triangle (a special right triangle with one angle measuring 30 degrees and another measuring 60 degrees), the sides have a specific ratio. The ratio of the sides in a 30-60-90 triangle is:

opposite side o: adjacent side a: hypotenuse h = √3: 1: 2

Using this ratio, we can determine the values for the sides of our triangle with the 60-degree angle. In this case, the opposite side “o” is √3 times the adjacent side “a” and the hypotenuse “h” is twice the adjacent side “a”. Therefore, we have:

o = √3 * a
h = 2 * a

For simplicity, let’s assume that the adjacent side “a” has a length of 1. Then:
o = √3 * 1 = √3
h = 2 * 1 = 2

Now we can substitute these values into the sine formula:
sin(60) = √3 / 2

Hence, sin(60) is equal to √3 / 2.

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