1. What is an Exact Value?

An exact value is a precise mathematical expression for a trigonometric function at specific angles, typically expressed as fractions or roots rather than decimal approximations.

2. How Does the Calculator Work?

The calculator uses predefined exact values for common angles:

Common Exact Values:

• \( \sin 0° = 0 \)
• \( \sin 30° = \frac{1}{2} \)
• \( \sin 45° = \frac{\sqrt{2}}{2} \)
• \( \sin 60° = \frac{\sqrt{3}}{2} \)
• \( \sin 90° = 1 \)

3. Importance of Exact Values

Details: Exact values are crucial in mathematics for precise calculations, proofs, and when working with symbolic expressions where decimal approximations would introduce errors.

4. Using the Calculator

Tips: Select an angle (0°, 30°, 45°, 60°, 90°) and a trigonometric function to see its exact value. The calculator will display the exact fractional or radical form.

5. Frequently Asked Questions (FAQ)

Q1: Why use exact values instead of decimals?
A: Exact values maintain mathematical precision and are often required in proofs, symbolic calculations, and exact solutions.

Q2: What angles have exact values?
A: Common angles (0°, 30°, 45°, 60°, 90° and their multiples) have known exact values based on special triangles.

Q3: How are exact values derived?
A: They come from geometric relationships in special triangles (30-60-90 and 45-45-90) and the unit circle.

Q4: Can I get exact values for any angle?
A: Only certain angles have simple exact forms. Most angles require decimal approximations or infinite series.

Q5: Are exact values used in real-world applications?
A: Yes, especially in engineering, physics, and computer graphics where precision is critical.