Six Trigonometric Functions Calculator
Calculate Trigonometric Functions
Enter an angle below to find its sine, cosine, tangent, cosecant, secant, and cotangent.
What are the Six Trigonometric Functions?
The Six Trigonometric Functions are fundamental functions in mathematics that relate the angles of a right-angled triangle to the ratios of the lengths of its sides. These functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). They are also defined using the unit circle, which allows them to be applied to angles beyond those in a right triangle (0 to 90 degrees or 0 to π/2 radians).
These functions are widely used in various fields such as geometry, calculus, physics, engineering, navigation, and even music theory. Anyone studying these subjects or working in these areas will frequently use the Six Trigonometric Functions.
A common misconception is that these functions only apply to right-angled triangles. While their initial definition comes from right triangles (SOH CAH TOA), their extension via the unit circle makes them applicable to any angle, positive or negative.
Six Trigonometric Functions Formula and Mathematical Explanation
For an angle θ within a right-angled triangle:
- Sine (sin θ) = Opposite / Hypotenuse
- Cosine (cos θ) = Adjacent / Hypotenuse
- Tangent (tan θ) = Opposite / Adjacent
- Cosecant (csc θ) = Hypotenuse / Opposite = 1 / sin θ
- Secant (sec θ) = Hypotenuse / Adjacent = 1 / cos θ
- Cotangent (cot θ) = Adjacent / Opposite = 1 / tan θ
On the unit circle (a circle with radius 1 centered at the origin), if we have an angle θ measured counterclockwise from the positive x-axis, the point where the terminal side of the angle intersects the circle has coordinates (x, y). Then:
- sin θ = y
- cos θ = x
- tan θ = y/x
- csc θ = 1/y (undefined when y=0)
- sec θ = 1/x (undefined when x=0)
- cot θ = x/y (undefined when y=0)
Variables Table
| Variable | Meaning | Unit | Typical Range (in unit circle context) |
|---|---|---|---|
| θ (theta) | The angle | Degrees or Radians | Any real number |
| Opposite | Length of the side opposite to angle θ in a right triangle | Length units | Positive |
| Adjacent | Length of the side adjacent to angle θ in a right triangle | Length units | Positive |
| Hypotenuse | Length of the hypotenuse in a right triangle | Length units | Positive, > Opposite, > Adjacent |
| x | x-coordinate on the unit circle | Dimensionless | -1 to 1 |
| y | y-coordinate on the unit circle | Dimensionless | -1 to 1 |
Practical Examples (Real-World Use Cases)
Example 1: Finding the Height of a Tree
You are standing 50 meters away from the base of a tree. You measure the angle of elevation from your eye level to the top of the tree to be 30 degrees. If your eye level is 1.5 meters above the ground, how tall is the tree?
Here, the distance to the tree is the adjacent side (50m), and the height of the tree above eye level is the opposite side. We use tan(30°) = Opposite / 50.
Opposite = 50 * tan(30°) ≈ 50 * 0.57735 = 28.8675 meters.
Total height of the tree = 28.8675 + 1.5 = 30.3675 meters.
Example 2: Simple Harmonic Motion
The displacement of an object in simple harmonic motion can be described by x(t) = A cos(ωt + φ). If an object has an amplitude A=0.5m, angular frequency ω=π rad/s, and phase φ=0, its position at t=0.5s is x(0.5) = 0.5 * cos(π*0.5) = 0.5 * cos(π/2) = 0.5 * 0 = 0 meters.
The Six Trigonometric Functions are crucial for analyzing wave phenomena and oscillations.
How to Use This Six Trigonometric Functions Calculator
- Enter the Angle Value: Type the angle into the "Angle Value" input field.
- Select the Unit: Choose whether the angle you entered is in "Degrees" or "Radians" using the radio buttons.
- Calculate (or Auto-Update): The results for all Six Trigonometric Functions (sin, cos, tan, csc, sec, cot) will be displayed automatically or after clicking "Calculate", along with the angle in radians if you entered it in degrees. The unit circle diagram will also update.
- Read the Results: The primary result section highlights key values, while the intermediate results list all six function values. Note that csc, sec, and cot may be "Undefined" for certain angles.
- Reset: Click "Reset" to clear the input and results and return to default values.
- Copy Results: Click "Copy Results" to copy the angle and the calculated trigonometric values to your clipboard.
Key Factors That Affect Six Trigonometric Functions Results
- Angle Value: The magnitude of the angle directly determines the values of the functions.
- Angle Unit: Whether the angle is measured in degrees or radians is crucial. The calculator converts to radians for the underlying `Math` functions. 180 degrees = π radians.
- Quadrant of the Angle: The signs (+ or -) of the trigonometric functions depend on which quadrant the terminal side of the angle lies in (I: All+, II: Sin+, III: Tan+, IV: Cos+).
- Proximity to Axes: Angles near 0, 90, 180, 270, 360 degrees (0, π/2, π, 3π/2, 2π radians) often result in values of 0, 1, -1, or undefined for some functions.
- Calculator Precision: The precision of the `Math` functions in JavaScript affects the number of decimal places in the result. Very small numbers near zero might cause large numbers or "Infinity" for reciprocals.
- Right Triangle Ratios (for angles 0-90): The relative lengths of opposite, adjacent, and hypotenuse define the functions within a right triangle.
Frequently Asked Questions (FAQ)
- What are radians?
- Radians are a unit of angle measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. π radians = 180 degrees.
- What is the unit circle?
- The unit circle is a circle with a radius of 1 centered at the origin (0,0) of a Cartesian coordinate system. It's used to define the Six Trigonometric Functions for all real-numbered angles.
- When are tan, csc, sec, cot undefined?
- Tan(θ) and sec(θ) are undefined when cos(θ)=0 (θ = 90°, 270°, …). Csc(θ) and cot(θ) are undefined when sin(θ)=0 (θ = 0°, 180°, 360°, …).
- How are the Six Trigonometric Functions used in real life?
- They are used in navigation (GPS, astronomy), engineering (building bridges, electronics), physics (waves, optics, mechanics), computer graphics, and many other fields to model periodic phenomena and geometric relationships.
- Can I enter negative angles?
- Yes, the calculator accepts negative angles. A negative angle is measured clockwise from the positive x-axis.
- What is SOH CAH TOA?
- SOH CAH TOA is a mnemonic to remember the definitions for sine, cosine, and tangent in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Why are they called "functions"?
- Because for each valid input angle, there is exactly one output value (or undefined) for each of the Six Trigonometric Functions.
- Are there other trigonometric functions?
- The six mentioned (sin, cos, tan, csc, sec, cot) are the primary ones. There are others like versine, haversine, exsecant, and excosecant, which are less common today but were historically important in navigation.
Related Tools and Internal Resources
- Right Triangle Calculator: Calculate sides and angles of a right triangle.
- Angle Converter: Convert between degrees, radians, and other angle units.
- Pythagorean Theorem Calculator: Find the missing side of a right triangle.
- Law of Sines Calculator: Solve non-right triangles using the Law of Sines.
- Law of Cosines Calculator: Solve non-right triangles using the Law of Cosines.
- Unit Circle Explorer: An interactive tool to explore the unit circle and trigonometric values.