📐 Trigonometry Calculator
Calculate sin, cos, tan, and inverse functions with degree/radian conversion
Sine (sin)
Cosine (cos)
Tangent (tan)
Arcsine (sin⁻¹)
Arccosine (cos⁻¹)
Arctangent (tan⁻¹)
Degrees → Radians
Radians → Degrees
📊 Trigonometric Functions Graph
Visual representation of sin(x), cos(x), and tan(x) functions
How to Use This Calculator
- Choose the trigonometric function you need: sin, cos, tan for forward calculations, or arcsin, arccos, arctan to find angles from ratios
- For sin, cos, and tan: select whether your angle is in degrees or radians using the dropdown menu
- Enter your angle value (for forward functions) or ratio value (for inverse functions)
- For inverse functions (arcsin, arccos, arctan), enter a value and get the angle in both degrees and radians
- Remember: arcsin and arccos only accept values between -1 and 1 (the calculator will show "Invalid" otherwise)
- Arctan accepts any real number input
- Use the degree/radian converters at the bottom to switch between angle units
- Results calculate automatically as you type with 4 decimal places precision
- Common angles: 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2) have exact known values
Understanding Trigonometry
What is Trigonometry?
Trigonometry is the branch of mathematics that studies relationships between angles and side lengths in triangles, especially right triangles. The word comes from Greek: "trigonon" (triangle) + "metron" (measure). The three primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—originated from studying right triangles but extend to any angle using the unit circle. In a right triangle with angle θ, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. For example, in a right triangle with a 30° angle, opposite side 1 unit, hypotenuse 2 units: sin(30°) = 1/2 = 0.5. Beyond triangles, trig functions model periodic phenomena like sound waves, light waves, tides, seasons, and circular motion. Trigonometry is essential in physics, engineering, astronomy, navigation, architecture, computer graphics, and signal processing. Students encounter it in Geometry, Algebra 2, Precalculus, and Calculus courses.
The Unit Circle and Special Angles
The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. Any point on the unit circle can be represented as (cos(θ), sin(θ)), where θ is the angle from the positive x-axis. This visualization extends trigonometry beyond right triangles to all angles (including negative and greater than 90°). Special angles have exact, memorable values. At 0°: sin = 0, cos = 1, tan = 0. At 30° (π/6 radians): sin = 1/2, cos = √3/2, tan = 1/√3. At 45° (π/4): sin = cos = √2/2, tan = 1. At 60° (π/3): sin = √3/2, cos = 1/2, tan = √3. At 90° (π/2): sin = 1, cos = 0, tan = undefined. These come from 30-60-90 triangles (side ratio 1:√3:2) and 45-45-90 triangles (side ratio 1:1:√2). Memorizing these values makes homework and exams much faster. The unit circle also shows that sin and cos repeat every 360° (2π radians)—this is called periodicity.
Inverse Trigonometric Functions
Inverse trig functions reverse the process: given a ratio, find the angle. Notation: arcsin (or sin⁻¹), arccos (or cos⁻¹), arctan (or tan⁻¹). Note: sin⁻¹ does NOT mean 1/sin; it means "inverse sine." If sin(30°) = 0.5, then arcsin(0.5) = 30°. Domain restrictions are crucial. Since sine and cosine values range from -1 to +1, arcsin and arccos only accept inputs in [-1, 1]. Entering arcsin(2) is invalid—there's no angle whose sine is 2. Arctan accepts any real number because tangent can output any value. Range (output) restrictions: arcsin returns angles in [-90°, 90°] or [-π/2, π/2], arccos returns [0°, 180°] or [0, π], arctan returns (-90°, 90°) or (-π/2, π/2). Example: arctan(1) = 45° because tan(45°) = 1. These functions are essential for solving triangles when you know sides but need angles, such as in navigation (finding bearing angles), engineering (calculating slopes), and physics (projectile angles).
Real-World Applications
Trigonometry is everywhere in science and engineering. Navigation: Pilots and sailors use trig to calculate bearings, distances, and course corrections. GPS systems rely on triangulation (using angles to find position). Physics: Projectile motion, wave interference, pendulum periods, and force components all use sin/cos/tan. Architecture & Construction: Roof pitches, ramp angles, and staircase designs require trig. Example: a roof with 6/12 pitch has angle arctan(6/12) ≈ 26.57°. Astronomy: Measuring star distances using parallax angles. Music & Sound: Sound waves are sine waves; Fourier analysis decomposes complex sounds into trig functions. Computer Graphics: Rotating objects, camera angles, and lighting calculations use rotation matrices built from sin/cos. Surveying: Measuring land areas and elevations using angle measurements from known points. Medicine: EKG and brain wave patterns analyzed as periodic functions. Sports: Optimal launch angles for projectiles (basketballs, golf balls) calculated with trig. Understanding trigonometry unlocks quantitative reasoning in countless technical fields.
Frequently Asked Questions
What are sine, cosine, and tangent?
Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions that relate angles to side ratios in right triangles. For an angle θ in a right triangle: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. Example: In a right triangle with angle 30°, opposite side 1, hypotenuse 2, then sin(30°) = 1/2 = 0.5. These functions extend beyond triangles to model waves, circles, and periodic phenomena.
What's the difference between degrees and radians?
Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians. Degrees divide a circle into 360 equal parts (historically from Babylonian astronomy). Radians measure angles based on circle radius: one radian is the angle where the arc length equals the radius. To convert: degrees × (π/180) = radians, and radians × (180/π) = degrees. Example: 90° = π/2 radians ≈ 1.5708 radians. Calculus and advanced math prefer radians.
What are inverse trig functions (arcsin, arccos, arctan)?
Inverse trig functions reverse the process: they take a ratio and return the angle. If sin(30°) = 0.5, then arcsin(0.5) = 30°. Domain restrictions: arcsin and arccos accept inputs between -1 and 1 only (since sin and cos can't exceed ±1). Arctan accepts any number. Example: If tan(θ) = 1, then θ = arctan(1) = 45° (or π/4 radians). These are written as sin⁻¹, cos⁻¹, tan⁻¹ or arcsin, arccos, arctan.
What angles are most common in trigonometry?
The special angles 0°, 30°, 45°, 60°, and 90° (or in radians: 0, π/6, π/4, π/3, π/2) appear constantly because their trig values are exact. Examples: sin(30°) = 1/2, sin(45°) = √2/2 ≈ 0.7071, sin(60°) = √3/2 ≈ 0.8660, cos(45°) = √2/2, tan(45°) = 1. Memorizing these values speeds up problem-solving in geometry, physics, and calculus. They come from 30-60-90 and 45-45-90 triangles.
Why is tan(90°) undefined?
Tangent is defined as tan(θ) = sin(θ)/cos(θ) = opposite/adjacent. At 90°, the adjacent side is zero (you're pointing straight up), and dividing by zero is undefined. Mathematically, sin(90°) = 1 and cos(90°) = 0, so tan(90°) = 1/0 = undefined. Similarly, tan is undefined at 270°, 450°, etc. (any odd multiple of 90°). The calculator will show 'Undefined' for these angles.
How accurate is this trig calculator?
This calculator uses JavaScript's built-in Math functions with double-precision floating-point arithmetic (about 15-17 significant digits). Results are displayed to 4 decimal places for readability. For homework, engineering, and most scientific applications, this precision is more than sufficient. Minor rounding may occur at extreme precision levels.
Is this trigonometry calculator free?
Yes! Completely free with no limitations, hidden fees, or required account. Use it for homework, exams, projects, or professional work unlimited times. All calculations run locally in your browser for privacy.