Understanding the Derivation of ( sin 60^circ frac{sqrt{3}}{2} ) in Trigonometry

The sine function is a fundamental concept in trigonometry, representing the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. The value ( sin 60^circ frac{sqrt{3}}{2} ) is a well-known result, derived from the properties of specific triangles. Let's explore this derivation step-by-step, using both a 30-60-90 triangle and an equilateral triangle.

Deriving ( sin 60^circ ) Using a 30-60-90 Triangle

In a 30-60-90 triangle, the sides are in a specific ratio:

The side opposite the ( 30^circ ) angle is the shortest side, typically denoted as 1. The side opposite the ( 60^circ ) angle is ( sqrt{3} ). The hypotenuse, which is the longest side, is 2.

The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. For ( 60^circ ):

[ sin 60^circ frac{text{opposite}}{text{hypotenuse}} frac{sqrt{3}}{2} ]

This derivation is a direct application of the properties of the 30-60-90 triangle. Thus, ( sin 60^circ frac{sqrt{3}}{2} ) is a fundamental result in trigonometry.

Deriving ( sin 60^circ ) Using an Equilateral Triangle

To derive ( sin 60^circ frac{sqrt{3}}{2} ) using an equilateral triangle, let's follow these steps:

Consider an equilateral triangle ( ABC ) with each angle equal to ( 60^circ ) and each side of length ( a ). Draw an altitude ( AD ) from vertex ( A ) to the midpoint ( D ) of side ( BC ). Since ( AD ) is the altitude, it divides the equilateral triangle into two congruent right-angled triangles ( ABD ) and ( ACD ). The angle ( angle ADC 90^circ ) (since it is a right triangle). The angle ( angle ACD 60^circ ) (since the original triangle is equilateral).

In one of the right-angled triangles, say ( ABD ), the side opposite the ( 60^circ ) angle is ( BD ), and the hypotenuse is the side of the equilateral triangle, which is ( a ).

The length of the altitude ( AD ) can be determined using the Pythagorean theorem:

[ AD^2 BD^2 AB^2 ]

Since ( BD frac{a}{2} ):

[ AD^2 left( frac{a}{2} right)^2 a^2 ]

[ AD^2 a^2 - frac{a^2}{4} frac{4a^2 - a^2}{4} frac{3a^2}{4} ]

[ AD sqrt{frac{3a^2}{4}} frac{sqrt{3}a}{2} ]

Now, the sine of ( 60^circ ) is the ratio of the opposite side (altitude) to the hypotenuse:

[ sin 60^circ frac{text{opposite}}{text{hypotenuse}} frac{frac{sqrt{3}a}{2}}{a} frac{sqrt{3}}{2} ]

Conclusion

In conclusion, the value ( sin 60^circ frac{sqrt{3}}{2} ) can be derived from the properties of both a 30-60-90 triangle and an equilateral triangle. These derivations provide a clear understanding of the trigonometric functions and their applications.

Related Keywords:

sin60 30-60-90 Triangle Equilateral Triangle