Six Trigonometric Functions Calculator
The six trigonometric functions calculator evaluates all six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) from multiple input methods including angles, coordinate points, and right triangle sides. This comprehensive tool provides exact values, decimal approximations, quadrant information, and reference angles for complete trigonometric analysis.
Select Input Method
Enter the coordinates of a point (x, y) on the terminal side of an angle in standard position:
Enter the sides of a right triangle (at least two values):
Results:
Understanding the Six Trigonometric Functions
The six trigonometric functions are fundamental mathematical functions that relate angles to ratios of sides in right triangles or coordinates on the unit circle. These functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot), each providing unique relationships essential for solving problems in geometry, physics, engineering, and many other fields.
Basic Definitions
For a right triangle with an acute angle θ, the six trigonometric functions are defined using three sides: the opposite side, adjacent side, and hypotenuse. The sine function equals the ratio of the opposite side to the hypotenuse, the cosine equals the adjacent side to the hypotenuse, and the tangent equals the opposite to the adjacent side.
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent
csc(θ) = Hypotenuse / Opposite = 1/sin(θ)
sec(θ) = Hypotenuse / Adjacent = 1/cos(θ)
cot(θ) = Adjacent / Opposite = 1/tan(θ)
Coordinate Definition
When using a point (x, y) on the terminal side of an angle in standard position, where r = √(x² + y²) is the distance from the origin, the trigonometric functions are calculated as ratios of these coordinates. This coordinate approach extends the definitions beyond acute angles to all real angles.
sin(θ) = y / r
cos(θ) = x / r
tan(θ) = y / x (where x ≠ 0)
csc(θ) = r / y (where y ≠ 0)
sec(θ) = r / x (where x ≠ 0)
cot(θ) = x / y (where y ≠ 0)
Standard Angle Values
Certain angles have exact trigonometric values that are frequently used in mathematics and should be memorized for quick reference. The table below shows the six trigonometric function values for the most common standard angles.
| Angle (Degrees) | Angle (Radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
Signs in Four Quadrants
The sign (positive or negative) of each trigonometric function depends on which quadrant the terminal side of the angle lies in. Understanding these sign patterns is crucial for evaluating trigonometric functions correctly for any angle.
| Function | Quadrant I (0° to 90°) | Quadrant II (90° to 180°) | Quadrant III (180° to 270°) | Quadrant IV (270° to 360°) |
|---|---|---|---|---|
| sin θ | Positive (+) | Positive (+) | Negative (−) | Negative (−) |
| cos θ | Positive (+) | Negative (−) | Negative (−) | Positive (+) |
| tan θ | Positive (+) | Negative (−) | Positive (+) | Negative (−) |
| csc θ | Positive (+) | Positive (+) | Negative (−) | Negative (−) |
| sec θ | Positive (+) | Negative (−) | Negative (−) | Positive (+) |
| cot θ | Positive (+) | Negative (−) | Positive (+) | Negative (−) |
How to Use This Calculator
Method 1: Angle Input
Select the "Angle Input" tab and enter any angle value in either degrees or radians. The calculator evaluates all six trigonometric functions for that angle, provides the reference angle, identifies the quadrant, and shows both exact values (when possible) and decimal approximations.
Method 2: Point Coordinates
If you have a point (x, y) on the terminal side of an angle in standard position, select the "Point Coordinates" tab. Enter the x and y coordinates, and the calculator computes the distance r from the origin, determines all six trigonometric functions using the coordinate definitions, calculates the actual angle, and identifies the quadrant.
Method 3: Right Triangle Sides
For right triangle problems, select the "Right Triangle" tab and enter at least two of the three sides (opposite, adjacent, or hypotenuse). The calculator uses the Pythagorean theorem to find the missing side if needed, then evaluates all six trigonometric functions for the angle formed by the given sides.
Reference Angles
A reference angle is the acute angle (between 0° and 90°) formed by the terminal side of an angle and the x-axis. Reference angles are useful because they allow you to find trigonometric values for any angle by relating it to an equivalent acute angle with known values.
• Quadrant I: Reference angle = θ
• Quadrant II: Reference angle = 180° − θ
• Quadrant III: Reference angle = θ − 180°
• Quadrant IV: Reference angle = 360° − θ
Reciprocal Relationships
Three of the six trigonometric functions are reciprocals of the other three. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. These relationships are fundamental to understanding how the functions relate to each other.
| Function | Reciprocal Function | Relationship |
|---|---|---|
| sin θ | csc θ | csc θ = 1 / sin θ |
| cos θ | sec θ | sec θ = 1 / cos θ |
| tan θ | cot θ | cot θ = 1 / tan θ |
Pythagorean Identities
The Pythagorean identities are fundamental relationships among the trigonometric functions derived from the Pythagorean theorem. These identities are essential for simplifying trigonometric expressions and solving trigonometric equations.
sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)
Common Applications
Trigonometric functions have widespread applications across numerous fields. In physics and engineering, they model periodic phenomena like waves, oscillations, and alternating currents. Navigation systems use trigonometry for calculating distances and bearings. Architecture and construction rely on these functions for designing structures and calculating angles. Computer graphics use trigonometric functions for rotations, transformations, and rendering three-dimensional objects on two-dimensional screens.
Real-World Examples
Engineers use trigonometric functions to calculate forces in structural analysis, determining stress and load distributions in buildings and bridges. Astronomers apply these functions to calculate distances to celestial objects and predict planetary positions. Signal processing uses trigonometric functions extensively in Fourier analysis to decompose complex waveforms into simpler components. Surveyors employ trigonometry to measure land areas and elevations without direct measurement.
Special Cases and Undefined Values
Certain angle values result in undefined trigonometric functions because they involve division by zero. Tangent and secant are undefined when cosine equals zero (at 90°, 270°, etc.). Cotangent and cosecant are undefined when sine equals zero (at 0°, 180°, 360°, etc.). Understanding these undefined cases is crucial for correctly interpreting trigonometric calculations.
Tips for Using Trigonometric Functions
Always verify which quadrant your angle lies in before determining the sign of the function. Use reference angles to simplify calculations by reducing any angle to an equivalent acute angle. Remember the reciprocal relationships to quickly convert between functions. When working with right triangles, clearly label the sides relative to the angle you're considering. For coordinate-based problems, always calculate the distance r first before evaluating any trigonometric function.
Frequently Asked Questions
What are the six trigonometric functions?
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). They represent relationships between angles and side ratios in right triangles or coordinates on the unit circle.
How do you find trigonometric functions given a point?
Given a point (x, y), first calculate r = √(x² + y²). Then use the coordinate definitions: sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, and the reciprocals for csc, sec, and cot. The calculator automates these calculations for you.
What is the ASTC rule for trigonometric signs?
ASTC stands for "All Students Take Calculus" and indicates which trigonometric functions are positive in each quadrant: All functions in Quadrant I, only Sine in Quadrant II, only Tangent in Quadrant III, and only Cosine in Quadrant IV.
When are trigonometric functions undefined?
Trigonometric functions become undefined when they involve division by zero. Tangent and secant are undefined when cos(θ) = 0 (at 90°, 270°, etc.), while cotangent and cosecant are undefined when sin(θ) = 0 (at 0°, 180°, 360°, etc.).
How do you evaluate trigonometric functions for any angle?
First, reduce the angle to a coterminal angle between 0° and 360° by adding or subtracting 360°. Determine which quadrant the angle is in, find the reference angle, evaluate the function for the reference angle, and then apply the appropriate sign based on the quadrant.
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