What is the exact value of sin 60 degrees, and how is it expressed as a simplified fraction?
The Exact Value of Sin 60°
In trigonometry, the sine of an angle represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. For an angle of 60 degrees, the exact value is √3/2.
Understanding the Geometry
To derive this value, we look at an equilateral triangle where all sides have a length of 2. If we draw an altitude from one vertex to the opposite side, we bisect the triangle into two 30-60-90 right triangles. The base of this new triangle is 1, the hypotenuse is 2, and the height (the altitude) is √3 (calculated via the Pythagorean theorem: 1² + h² = 2²).
Since sine is defined as opposite / hypotenuse, we take the side opposite the 60° angle (which is √3) and divide it by the hypotenuse (which is 2). Thus, the exact value is √3/2.
Quick Reference Table
Below is a comparison of the primary trigonometric ratios for the 60-degree angle:
| Function | Exact Value |
|---|---|
| sin 60° | √3/2 |
| cos 60° | 1/2 |
| tan 60° | √3 |
Real-World Examples
Understanding these exact values is crucial in fields like engineering, physics, and computer graphics. For instance, when calculating the forces acting on a structural beam at an angle or determining the coordinates of a vertex in a 3D game engine, using exact values like √3/2 ensures precision that decimal approximations (like 0.866) cannot provide.
Common Pitfalls
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Degrees vs. Radians: Always ensure your calculator or mental model is set to the correct unit. Sin(60) in radians is a completely different value than sin(60°) in degrees.
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Decimal Approximation: While 0.866 is a common approximation, it is not the exact value. If a test or problem asks for a "simplified fraction" or "exact value," you must use the radical form (√3/2).
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Confusing Sine and Cosine: A common mistake is swapping the values for 30° and 60°. Remember: sin 60° is the "larger" value (√3/2 ≈ 0.866), while sin 30° is the "smaller" value (1/2 = 0.5).