三角恒等式
基本三角恒等式及其应用
Interactive Trigonometric Identities Visualization
Welcome to the Trigonometric Identities Explorer
This interactive visualization helps you understand and verify trigonometric identities by showing that both sides of an identity are equal for any angle. Explore different identities and see how they hold true.
What you can explore:
- Pythagorean Identity - sin²(θ) + cos²(θ) = 1
- Angle Sum Identity - sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
- Angle Difference Identity - sin(α - β) = sin(α)cos(β) - cos(α)sin(β)
- Double Angle Identity - sin(2θ) = 2sin(θ)cos(θ)
- Half Angle Identity - sin²(θ/2) = (1 - cos(θ))/2
- Product-to-Sum Identity - sin(α)sin(β) = [cos(α-β) - cos(α+β)]/2
How to Use This Visualization
Interactive Features:
- • Select Identity Type - Choose from 6 different trigonometric identities
- • Adjust Angles - Change angle values to test the identity
- • Toggle Graph - Show/hide the graphical representation
- • Verify Equality - See real-time calculation of both sides
What You'll See:
- • Left Side (green, solid) - The left side of the identity
- • Right Side (blue, dashed) - The right side of the identity
- • Verification - Real-time check showing if both sides are equal
- • Values Display - Exact values of both sides with difference
Fundamental identity relating sine and cosine
Identity Verification
Identity: sin²(θ) + cos²(θ) = 1
Left Side:
1.000000
Right Side:
1.000000
✓ Identity Verified! Both sides are equal.
The identity holds true for the selected angle(s).
Current Values
Angle α (θ): 30° (0.5236 rad)
Left Side: 1.000000
Right Side: 1.000000
Key Concepts
Identity: An equation that is true for all values of the variable(s).
Verification: Both sides of an identity should produce the same value for any angle.
Applications: Identities are used to simplify expressions, solve equations, and prove other mathematical results.
Common Trigonometric Identities
Pythagorean:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Angle Sum/Difference:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
Double Angle:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
Half Angle:
- sin²(θ/2) = (1 - cos(θ))/2
- cos²(θ/2) = (1 + cos(θ))/2
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