Trigonometry Calculator.
Explore sine, cosine, tangent and their inverses with interactive unit circle visualization. Convert between degrees and radians with precision.
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Step-by-Step Solution
See exactly how the trigonometric calculation works
Understanding the Calculation
Sine of 45 degrees represents the y-coordinate of a point on the unit circle at a 45-degree angle from the positive x-axis.
Step 1: Convert angle to radians: 45° × π/180 = π/4 ≈ 0.7854 radians
Step 2: Calculate sin(π/4) using the unit circle or Taylor series
Result: sin(45°) = √2/2 ≈ 0.7071
Smart Insights
Discover useful facts about trigonometric functions
Unit Circle Connection
On the unit circle, the x-coordinate equals cos(θ) and the y-coordinate equals sin(θ). This is the foundation of all trigonometry.
Common Angles
sin(30°) = 0.5, sin(45°) = 0.707, sin(60°) = 0.866. Memorizing these helps with quick mental math.
Periodic Nature
Trig functions repeat every 360° (2π radians). sin(θ) = sin(θ + 360°). This makes them perfect for modeling waves and cycles.
Real-World Applications
Trigonometry is used in GPS navigation, sound waves, light waves, engineering, architecture, and even music synthesis.
Understanding Trigonometry
What is Trigonometry?
Trigonometry studies the relationships between angles and sides of triangles. The word comes from Greek: "trigonon" (triangle) and "metron" (measure). It's essential for understanding waves, circles, and periodic phenomena.
The Three Main Functions
- sin(θ) = Opposite / Hypotenuse — the y-coordinate on unit circle
- cos(θ) = Adjacent / Hypotenuse — the x-coordinate on unit circle
- tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ) — the slope
Degrees vs Radians
- 180° = π radians (the fundamental conversion)
- 1° = π/180 radians ≈ 0.01745 radians
- 1 radian = 180/π degrees ≈ 57.296°
- 360° = 2π radians (full circle)
Key Values to Remember
- sin(0°) = 0, sin(30°) = 0.5, sin(45°) ≈ 0.707
- sin(60°) ≈ 0.866, sin(90°) = 1
- cos(0°) = 1, cos(30°) ≈ 0.866, cos(45°) ≈ 0.707
- cos(60°) = 0.5, cos(90°) = 0
Inverse Trigonometric Functions
Inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) reverse the process: given a ratio, they find the angle. They're used in robotics, GPS, game physics, and anywhere you need to find an angle from coordinates.
Pythagorean Identity
The most important trig identity: sin²(θ) + cos²(θ) = 1. This comes directly from the Pythagorean theorem applied to the unit circle where the radius is always 1.
Frequently Asked Questions
Why is tan(90°) undefined?
tan(90°) = sin(90°)/cos(90°) = 1/0, and division by zero is undefined. On the unit circle, the tangent line becomes vertical at 90°, meaning it extends to infinity.
Can sin or cos values be greater than 1?
No. Since sin(θ) and cos(θ) represent coordinates on the unit circle (radius = 1), their values are always between -1 and 1. However, tan(θ) can be any real number.
What are inverse trig functions used for?
Inverse functions (arcsin, arccos, arctan) find angles when you know the ratio. For example, if you know the rise and run of a ramp, arctan gives you the angle of inclination.
How do I convert between degrees and radians?
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. For example, 45° = 45 × π/180 = π/4 radians.
Why are trig functions periodic?
Because angles repeat every 360° (2π radians), trig functions have the same values at θ and θ + 360°. This periodicity makes them ideal for modeling waves, sound, and oscillations.
Where is trigonometry used in real life?
Trigonometry is used in GPS navigation, satellite dishes, music production, game development, bridge engineering, sound wave analysis, and even in medical imaging like CT scans.