## sin 60°

### The sine of 60° is a trigonometric function that represents the ratio of the length of the side opposite to the angle 60° to the length of the hypotenuse in a right triangle

The sine of 60° is a trigonometric function that represents the ratio of the length of the side opposite to the angle 60° to the length of the hypotenuse in a right triangle.

To understand the value of sin 60°, we can consider a right triangle where the angle opposite the side of length 1 unit is 60°. Let’s call this side “opposite” (o) and the hypotenuse “hyp” (h).

By using the Pythagorean theorem, we can find the length of the other side (adjacent) (a):

a² + o² = h²

a² + 1² = h²

a² + 1 = h²

a² = h² – 1

Now, let’s find the value of h. Since it’s a right triangle and one angle is 90°, we know that h² = o² + a². Hence:

h² = (1)² + (h² – 1)

h² = 1 + h² – 1

h² – h² = 1 – 1

0 = 0

As we can see, h² and h² cancel each other out, so we are left with 0 = 0 which holds true. This means that h can be any value.

Now, let’s specifically consider the case where h = 2 units. Plugging this value into the equation, we can solve for a:

a² = (2)² – 1

a² = 4 – 1

a² = 3

a = √3

Therefore, in the right triangle with an angle of 60°, the side opposite to the angle is √3 (approximately 1.73) and the hypotenuse is 2.

Now, coming back to the definition of sine, it is the ratio of the length of the side opposite the angle to the length of the hypotenuse:

sin 60° = opposite/hypotenuse

sin 60° = √3/2

sin 60° is approximately 0.866.

Therefore, the value of sin 60° is 0.866.

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