Find the exact value of sin 60 degree
The correct answer and explanation is :
The exact value of sin 60° is:
$$
\sin 60^\circ = \frac{\sqrt{3}}{2}
$$
🔍 Explanation (300+ words):
To understand why $\sin 60^\circ = \frac{\sqrt{3}}{2}$, let’s explore the geometry and trigonometry behind it.
🌟 Using a Special Triangle – The 30°-60°-90° Triangle:
A 30°-60°-90° triangle is a special right triangle where the angles are 30°, 60°, and 90°. In this triangle, the sides follow a consistent ratio:
- The side opposite 30° = 1 unit
- The side opposite 60° = $\sqrt{3}$ units
- The hypotenuse = 2 units
These ratios come from splitting an equilateral triangle in half. Here’s how:
- Start with an equilateral triangle (all sides equal, all angles 60°).
- Draw an altitude from one vertex to the opposite side. This divides the triangle into two right triangles.
- Now, each right triangle has angles of 30°, 60°, and 90°, and the original side of length 2 is split in half (so the base becomes 1).
- Using the Pythagorean theorem:
$$
\text{height} = \sqrt{2^2 – 1^2} = \sqrt{4 – 1} = \sqrt{3}
$$
So now, we have:
- Opposite side to 60° = $\sqrt{3}$
- Hypotenuse = 2
📐 Using the Definition of Sine:
By definition in trigonometry:
$$
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
$$
For $\theta = 60^\circ$, the opposite side is $\sqrt{3}$, and the hypotenuse is 2:
$$
\sin 60^\circ = \frac{\sqrt{3}}{2}
$$
✅ Verification with Unit Circle:
On the unit circle, the coordinates of a point at 60° (or $\frac{\pi}{3}$ radians) are:
$$
\left(\cos 60^\circ, \sin 60^\circ\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)
$$
Again, we see that:
$$
\sin 60^\circ = \frac{\sqrt{3}}{2}
$$
✅ Final Answer:
$$
\boxed{\frac{\sqrt{3}}{2}}
$$