Sin60 value

General
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sin60 value

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What is the value of sin(60 degrees)?

Answer:
The sine of 60^\circ is a commonly used trigonometric value. To find it, we can refer to the unit circle or use the properties of special triangles, specifically the 30-60-90 triangle.

Solution By Steps:

  1. Understanding a 30-60-90 Triangle:

    • A 30-60-90 triangle is a special right triangle where the angles are 30^\circ, 60^\circ, and 90^\circ.
    • The sides of a 30-60-90 triangle have a specific ratio: the length of the side opposite the 30^\circ angle is the shortest and is denoted as a, the side opposite the 60^\circ angle is \sqrt{3}a, and the hypotenuse is 2a.

  2. Using the Ratios to Determine \sin(60^\circ):

    • In a 30-60-90 triangle, \sin(\theta) is the ratio of the length of the side opposite the angle to the hypotenuse.

    • For \theta = 60^\circ, \sin(60^\circ) is therefore the ratio of the length of the side opposite the 60^\circ angle (\sqrt{3}a) to the hypotenuse (2a).

      \sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}a}{2a}

  3. Simplifying the Expression:

    • Simplify the fraction by dividing both the numerator and the denominator by a:

      \sin(60^\circ) = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2}

Final Answer:
The value of \sin(60^\circ) is:

\sin(60^\circ) = \frac{\sqrt{3}}{2}

The value \frac{\sqrt{3}}{2} is approximately:

\frac{\sqrt{3}}{2} \approx 0.86602540378
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What is the value \frac{\sqrt{3}}{2}

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What is the value \frac{\sqrt{3}}{2}?

Answer:
The value \frac{\sqrt{3}}{2} is an irrational number approximately equal to 0.86602540378. This value arises frequently in trigonometry, especially in the context of well-known angles such as 60^\circ or \frac{\pi}{3} radians.

Detailed Explanation:

  1. Approximate Decimal Value:

    • The square root of 3 (\sqrt{3}) is approximately 1.73205080757. Therefore, \frac{\sqrt{3}}{2} can be approximated by the decimal:

      \frac{\sqrt{3}}{2} \approx \frac{1.73205080757}{2} \approx 0.86602540378

  2. Significance in Trigonometry:

    • The value \frac{\sqrt{3}}{2} is significant in trigonometry as it represents:
      • \sin(60^\circ) or \sin\left(\frac{\pi}{3}\right)
      • \cos(30^\circ) or \cos\left(\frac{\pi}{6}\right)
    • These are fundamental values used in solving various trigonometric problems and in understanding the properties of angles in a unit circle.

  3. Use in Geometry:

    • This value also appears in geometrical shapes, particularly in equilateral triangles and other related configurations.

Interpretation and Practical Use:
Understanding the exact value (\frac{\sqrt{3}}{2}) and its decimal approximation (\approx 0.86602540378) is crucial in various mathematical calculations, including:

  • Solving trigonometric equations
  • Calculating component vectors in physics
  • Analyzing periodic functions like sine and cosine

Final Answer:
The value \frac{\sqrt{3}}{2} is approximately:

\frac{\sqrt{3}}{2} \approx 0.86602540378
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