Sin Cos Tan Calculator for Right Triangles
Welcome to the Sin Cos Tan Calculator. Input two side lengths of a right-angled triangle (opposite and adjacent to angle A), and we'll calculate the hypotenuse, angles, and the sine, cosine, and tangent values for both acute angles.
Right Triangle Details
Results
Hypotenuse (c): –
Angle A (θ): – degrees
Angle B (φ): – degrees
Sin(B): –
Cos(B): –
Tan(B): –
Formulas Used: c = √(a² + b²), A = atan(a/b), B = atan(b/a), sin(A)=a/c, cos(A)=b/c, tan(A)=a/b
Trigonometric Values for Angle A
Bar chart showing Sin(A), Cos(A), and Tan(A) values.
Summary Table
| Item | Value |
|---|---|
| Side a | – |
| Side b | – |
| Hypotenuse c | – |
| Angle A | – |
| Angle B | – |
| Sin(A) | – |
| Cos(A) | – |
| Tan(A) | – |
| Sin(B) | – |
| Cos(B) | – |
| Tan(B) | – |
Summary of side lengths, angles, and trigonometric values.
What is a Sin Cos Tan Calculator?
A Sin Cos Tan Calculator is a tool used to determine the values of the three primary trigonometric functions – sine (sin), cosine (cos), and tangent (tan) – for a given angle within a right-angled triangle. It can also work backward, finding angles if you know the ratio of sides. In the context of a triangle, these functions relate the angles of a right-angled triangle to the ratios of the lengths of its sides.
This specific Sin Cos Tan Calculator takes the lengths of the two shorter sides (opposite and adjacent to one of the acute angles) of a right-angled triangle and calculates the hypotenuse, the two acute angles, and the sin, cos, and tan values for both acute angles.
Who should use it? Students studying trigonometry, engineers, architects, game developers, and anyone needing to solve problems involving right-angled triangles and their angles will find this Sin Cos Tan Calculator useful.
Common Misconceptions: A common misconception is that sin, cos, and tan are lengths themselves; they are actually ratios of lengths, dimensionless numbers that depend on the angle.
Sin Cos Tan Formulas and Mathematical Explanation
For a right-angled triangle with sides 'a' (opposite angle A), 'b' (adjacent to angle A and opposite angle B), and 'c' (hypotenuse):
- Pythagorean Theorem: c² = a² + b² => c = √(a² + b²)
- Sine (sin): sin(A) = Opposite / Hypotenuse = a / c
- Cosine (cos): cos(A) = Adjacent / Hypotenuse = b / c
- Tangent (tan): tan(A) = Opposite / Adjacent = a / b
These are often remembered by the mnemonic SOH CAH TOA:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
To find the angles from the sides, we use the inverse trigonometric functions:
- Angle A = arctan(a/b) or arcsin(a/c) or arccos(b/c)
- Angle B = arctan(b/a) or arcsin(b/c) or arccos(a/c)
- Angle A + Angle B = 90 degrees (or π/2 radians)
Our Sin Cos Tan Calculator uses `atan(a/b)` to find angle A in radians, then converts it to degrees, and similarly for angle B.
| Variable | Meaning | Unit | Typical range |
|---|---|---|---|
| a | Length of side opposite angle A | Length units (e.g., cm, m, inches) | > 0 |
| b | Length of side adjacent to angle A | Length units | > 0 |
| c | Length of hypotenuse | Length units | > a, > b |
| A | Angle opposite side a | Degrees or Radians | 0° < A < 90° |
| B | Angle opposite side b | Degrees or Radians | 0° < B < 90° |
| sin(A), cos(A), tan(A) | Trigonometric ratios for angle A | Dimensionless | sin, cos: [-1, 1]; tan: (-∞, ∞) |
Practical Examples (Real-World Use Cases)
Example 1: Building a Ramp
Suppose you want to build a ramp that rises 1 meter (opposite side 'a') over a horizontal distance of 5 meters (adjacent side 'b'). You want to find the angle of the ramp and its length (hypotenuse).
- Opposite (a) = 1 m
- Adjacent (b) = 5 m
Using the Sin Cos Tan Calculator:
- Hypotenuse (c) ≈ 5.10 m
- Angle A ≈ 11.31°
- sin(A) ≈ 0.196, cos(A) ≈ 0.981, tan(A) = 0.2
The ramp will be about 5.1 meters long and make an angle of about 11.31 degrees with the ground.
Example 2: Navigation
A ship sails 10 km East (adjacent 'b') and then 7 km North (opposite 'a'). How far is it from the start, and what is its bearing from the starting point (relative to East)?
- Opposite (a) = 7 km
- Adjacent (b) = 10 km
Using the Sin Cos Tan Calculator:
- Hypotenuse (c) ≈ 12.21 km (distance from start)
- Angle A ≈ 35.00° (angle North of East)
- sin(A) ≈ 0.574, cos(A) ≈ 0.819, tan(A) = 0.7
The ship is about 12.21 km from the start, at a bearing of 35 degrees North of East.
How to Use This Sin Cos Tan Calculator
- Enter Side Lengths: Input the length of the side opposite angle A (side 'a') and the side adjacent to angle A (side 'b') into the respective fields.
- View Results: The calculator automatically updates and displays the hypotenuse 'c', angles A and B in degrees, and the sin, cos, and tan values for both angles A and B. The primary results (sin(A), cos(A), tan(A)) are highlighted.
- Analyze Chart and Table: The bar chart visually represents sin(A), cos(A), and tan(A). The table summarizes all input and calculated values.
- Copy Results: Use the "Copy Results" button to copy the key values to your clipboard.
- Reset: Click "Reset" to return to default input values.
Understanding the results from the Sin Cos Tan Calculator allows you to solve for unknown angles or side lengths in right-angled triangle problems.
Key Factors That Affect Sin Cos Tan Results
- Angle Magnitude: The values of sin, cos, and tan are entirely dependent on the angle itself (or the ratio of sides that define the angle). Small changes in the angle can lead to different ratio values.
- Ratio of Sides: Since sin, cos, and tan are ratios (a/c, b/c, a/b), changing the lengths of 'a' or 'b' while keeping the triangle right-angled will change the angles and thus the trig values.
- Units of Sides: While the trig values are dimensionless, ensure 'a' and 'b' are in the same units for the hypotenuse 'c' to be in the same unit and for the angle calculation to be correct. The Sin Cos Tan Calculator assumes consistent units.
- Quadrant (for general angles): Although this calculator focuses on right triangles (angles between 0° and 90°), for general angles, the quadrant determines the sign (+ or -) of sin, cos, and tan.
- Calculator Mode (Degrees/Radians): This calculator uses and displays angles in degrees, but the underlying `Math.atan` function returns radians, requiring conversion. Be mindful of units when using formulas elsewhere.
- Input Precision: The precision of your input side lengths will affect the precision of the calculated angles and trigonometric values from the Sin Cos Tan Calculator.
Frequently Asked Questions (FAQ)
- Q1: What is SOH CAH TOA?
- A1: SOH CAH TOA is a mnemonic to remember the definitions of sine, cosine, and tangent in a right-angled triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Q2: Can I use this calculator for non-right-angled triangles?
- A2: No, this specific Sin Cos Tan Calculator is designed for right-angled triangles using SOH CAH TOA. For non-right-angled triangles, you would use the Law of Sines or the Law of Cosines.
- Q3: What are the units of sin, cos, and tan?
- A3: Sine, cosine, and tangent are ratios of lengths, so they are dimensionless numbers – they have no units.
- Q4: Why does tan(90°) give an error or undefined?
- A4: In a right triangle context, as an angle approaches 90°, the adjacent side approaches 0. Tan(A) = Opposite/Adjacent, so division by zero occurs, making tan(90°) undefined. Our calculator focuses on angles derived from sides, so it won't directly calculate for 90° as an input angle derived this way.
- Q5: How do I find the angle if I know sin, cos, or tan?
- A5: You use the inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), or arctan (tan⁻¹). For example, if sin(A) = 0.5, then A = arcsin(0.5) = 30°.
- Q6: What happens if I enter negative side lengths?
- A6: Side lengths of a triangle must be positive. This Sin Cos Tan Calculator will show an error if you enter non-positive values for side lengths.
- Q7: What is the range of values for sin and cos?
- A7: For any angle, the values of sine and cosine range from -1 to +1, inclusive. For angles in a right triangle (0° to 90°), the range is 0 to 1.
- Q8: What is the range of values for tan?
- A8: The tangent value can range from -∞ to +∞. For angles between 0° and 90°, it ranges from 0 to +∞.
Related Tools and Internal Resources
- Pythagorean Theorem Calculator: Calculate the sides of a right triangle given two other sides.
- Angle Converter (Degrees/Radians): Convert angles between degrees and radians.
- Law of Sines Calculator: Solve non-right-angled triangles given certain angles and sides.
- Law of Cosines Calculator: Solve non-right-angled triangles given certain sides and an angle.
- Basic Geometry Formulas: A reference for common geometry formulas.
- Trigonometry Identities: Learn about fundamental trigonometric identities.