How to Find the Exact Value of sin 60°: Trigonometry with Equilateral Triangles

sin 60°

√3/ 2

The exact value of sin 60° is (√3) / 2.

To understand how to find this value, consider an equilateral triangle where all three sides are equal, and all three angles are also equal to 60°.

Now, draw an altitude from one of the vertices to the opposite side, which will divide the triangle into two congruent right triangles. The length of the altitude will be half the length of the base.

Let’s assume that the length of each side of the triangle is ‘a’. Therefore, the length of the base will also be ‘a’. The altitude will be ‘a/2’, and the hypotenuse of one of the right triangles will be ‘a’.

Now, we can use the Pythagorean theorem to find the length of the other leg of the triangle:

a² = (a/2)² + x² (where x is the length of the other leg)

Solving for x, we get:

x = √[(3a²) / 4]

Now, we can find the sine of 60° by using the definition of sine:

sin 60° = opposite / hypotenuse = x / a = (√3 / 2)

Therefore, the value of sin 60° is (√3) / 2.

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