Sine Cosine Tangent Calculator
Easily calculate the sine, cosine, and tangent values for any angle entered in degrees or radians using our Sine Cosine Tangent Calculator.
Trigonometric Function Calculator
Common Angles and Their Values
| Angle (Degrees) | Angle (Radians) | Sine (sin θ) | Cosine (cos θ) | Tangent (tan θ) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 or √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | Undefined |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | Undefined |
| 360° | 2π | 0 | 1 | 0 |
Table of sine, cosine, and tangent values for common angles.
Sine and Cosine Waves Visualization
Graph of y=sin(x) and y=cos(x) from 0 to 360 degrees, with the input angle marked.
What is a Sine Cosine Tangent Calculator?
A Sine Cosine Tangent Calculator is a tool designed to find the values of the three primary trigonometric functions – sine (sin), cosine (cos), and tangent (tan) – for a given angle. You input an angle, specify whether it's in degrees or radians, and the calculator instantly provides the sin, cos, and tan values. These functions are fundamental in trigonometry, relating the angles of a triangle to the lengths of its sides.
This calculator is useful for students learning trigonometry, engineers, scientists, architects, and anyone working with angles and their geometric relationships. It eliminates the need for manual calculations or looking up values in trigonometric tables. Our Sine Cosine Tangent Calculator also visualizes the sine and cosine waves, helping users understand the periodic nature of these functions.
Who should use it?
- Students studying mathematics, physics, or engineering.
- Teachers and educators explaining trigonometric concepts.
- Engineers and architects working with angles and dimensions.
- Scientists analyzing wave phenomena or other periodic data.
- Anyone needing quick and accurate trigonometric function values.
Common Misconceptions
A common misconception is that sine, cosine, and tangent only apply to right-angled triangles. While they are initially defined using right-angled triangles, their definitions are extended via the unit circle to apply to any angle, positive or negative, including angles greater than 360 degrees. Another point of confusion is the unit of the angle; always ensure you know whether you are working in degrees or radians, as the Sine Cosine Tangent Calculator requires this input.
Sine, Cosine, and Tangent Formula and Mathematical Explanation
The trigonometric functions sine, cosine, and tangent are based on the ratios of the sides of a right-angled triangle relative to one of its acute angles (θ).
- Sine (sin θ) = Length of the side Opposite to angle θ / Length of the Hypotenuse
- Cosine (cos θ) = Length of the side Adjacent to angle θ / Length of the Hypotenuse
- Tangent (tan θ) = Length of the side Opposite to angle θ / Length of the side Adjacent to angle θ = sin θ / cos θ
For a more general definition applicable to all angles, we use the unit circle (a circle with a radius of 1 centered at the origin of a Cartesian coordinate system). If we draw a line from the origin at an angle θ (measured counter-clockwise from the positive x-axis) to intersect the unit circle at a point (x, y), then:
- cos θ = x
- sin θ = y
- tan θ = y / x (undefined when x=0, i.e., at 90°, 270°, etc.)
Our Sine Cosine Tangent Calculator uses these definitions to compute the values.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| θ (Angle) | The input angle for which functions are calculated | Degrees or Radians | Any real number |
| Opposite | Length of the side opposite angle θ in a right triangle | Length units | Positive |
| Adjacent | Length of the side adjacent to angle θ in a right triangle (not hypotenuse) | Length units | Positive |
| Hypotenuse | Length of the side opposite the right angle (longest side) | Length units | Positive |
| sin θ | Sine of angle θ | Dimensionless ratio | -1 to 1 |
| cos θ | Cosine of angle θ | Dimensionless ratio | -1 to 1 |
| tan θ | Tangent of angle θ | Dimensionless ratio | -∞ to +∞ |
Practical Examples (Real-World Use Cases)
Example 1: Calculating Height
Suppose you are standing 50 meters away from the base of a tall building. You measure the angle of elevation from your eye level to the top of the building to be 60 degrees. If your eye level is 1.5 meters above the ground, how tall is the building?
Here, the distance from the building (50m) is the adjacent side, and the height of the building above your eye level is the opposite side relative to the 60-degree angle. We use the tangent function:
tan(60°) = Opposite / Adjacent
tan(60°) = Height_above_eye / 50m
Height_above_eye = 50m * tan(60°)
Using the Sine Cosine Tangent Calculator or knowing tan(60°) ≈ 1.732, Height_above_eye ≈ 50 * 1.732 = 86.6 meters.
Total height of the building = 86.6m + 1.5m = 88.1 meters.
Example 2: Navigation
A ship is sailing and its radar detects an obstacle at a distance of 5 nautical miles, at an angle of 30 degrees north of east (bearing 060°). How far north and how far east is the obstacle from the ship?
The distance (5 nautical miles) is the hypotenuse. The angle with the east direction is 30 degrees.
Distance East = Hypotenuse * cos(30°) = 5 * cos(30°) ≈ 5 * 0.866 = 4.33 nautical miles East.
Distance North = Hypotenuse * sin(30°) = 5 * sin(30°) = 5 * 0.5 = 2.5 nautical miles North.
Our Sine Cosine Tangent Calculator can quickly provide sin(30°) and cos(30°).
How to Use This Sine Cosine Tangent Calculator
- Enter the Angle: Type the numerical value of the angle into the "Angle" input field.
- Select the Unit: Choose whether the angle you entered is in "Degrees (°)" or "Radians (rad)" from the dropdown menu.
- Calculate: Click the "Calculate" button. The calculator will automatically process the input and display the results when you type or change the unit as well.
- View Results: The calculated values for sine, cosine, and tangent of the angle will be displayed in the "Results" section. You'll also see the angle converted to the other unit and the quadrant it falls in.
- See the Chart: The graph will update to show the sine and cosine waves, with a marker indicating your input angle and its corresponding sin and cos values.
- Reset: Click "Reset" to clear the input and results and return to default values (30 degrees).
- Copy: Click "Copy Results" to copy the main results and input to your clipboard.
The Sine Cosine Tangent Calculator is designed to be intuitive and fast.
Key Factors That Affect Sine, Cosine, and Tangent Results
- Angle Value: The primary factor is the magnitude of the angle itself. Different angles yield different ratios and thus different sin, cos, and tan values.
- Angle Unit (Degrees vs. Radians): The numerical value of an angle is different in degrees and radians (e.g., 90° = π/2 radians ≈ 1.57 rad). The Sine Cosine Tangent Calculator needs the correct unit to interpret the input value accurately.
- Quadrant: The quadrant in which the angle terminates (0-90°, 90-180°, 180-270°, 270-360°) determines the signs (+ or -) of the sine, cosine, and tangent values.
- Quadrant I (0-90°): All positive
- Quadrant II (90-180°): Sine positive, Cosine & Tangent negative
- Quadrant III (180-270°): Tangent positive, Sine & Cosine negative
- Quadrant IV (270-360°): Cosine positive, Sine & Tangent negative
- Reference Angle: For angles outside 0-90°, the trigonometric values are related to those of the reference angle (the acute angle formed with the x-axis).
- Periodicity: Trigonometric functions are periodic (sin and cos repeat every 360° or 2π radians, tan repeats every 180° or π radians). So, sin(θ) = sin(θ + 360°n) for any integer n.
- Calculator Precision: The precision of the calculator or software used can affect the number of decimal places in the result. Our Sine Cosine Tangent Calculator uses standard JavaScript Math functions.
Frequently Asked Questions (FAQ)
- What is sine (sin)?
- Sine is a trigonometric function that, for a given angle in a right-angled triangle, is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the unit circle, it's the y-coordinate.
- What is cosine (cos)?
- Cosine is a trigonometric function that, for a given angle in a right-angled triangle, is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In the unit circle, it's the x-coordinate.
- What is tangent (tan)?
- Tangent is a trigonometric function that, for a given angle in a right-angled triangle, is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. It's also equal to sin/cos.
- Why is tan(90°) undefined?
- At 90° (or π/2 radians), the adjacent side in a conceptual right triangle approaches zero, or the x-coordinate on the unit circle is 0. Since tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0, we get division by zero, which is undefined.
- How do I convert degrees to radians?
- To convert degrees to radians, multiply the angle in degrees by π/180. Our Sine Cosine Tangent Calculator can handle both units and shows the conversion.
- How do I convert radians to degrees?
- To convert radians to degrees, multiply the angle in radians by 180/π. Check out our Angle Measurement Converter for easy conversions.
- Can I use this Sine Cosine Tangent Calculator for negative angles?
- Yes, you can enter negative angle values. The calculator will correctly find the sin, cos, and tan values based on the unit circle definitions for negative angles (measured clockwise from the positive x-axis).
- Where can I learn more about the unit circle?
- The unit circle is fundamental to understanding trigonometry beyond right triangles. You might find our Unit Circle Calculator and explanations helpful.