What is Trigonometry?
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. The word comes from Greek: "trigonon" (triangle) + "metron" (measure).
Trigonometry is fundamental in many fields including physics, engineering, astronomy, navigation, and computer graphics.
The Six Trigonometric Functions
For an angle θ in a right triangle:
Primary Functions:
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
Reciprocal Functions:
csc(θ) = 1 / sin(θ) = hypotenuse / opposite
sec(θ) = 1 / cos(θ) = hypotenuse / adjacent
cot(θ) = 1 / tan(θ) = adjacent / opposite
Mnemonic: SOH-CAH-TOA
A popular way to remember the primary trig functions:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
Common Angle Values
| Angle | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Right Triangle Solver
In a right triangle, if you know any two values (sides or angles), you can find all the others using these relationships:
Pythagorean Theorem: a² + b² = c²
Where c is the hypotenuse, a and b are the other sides
Angle Sum: A + B = 90° (in a right triangle)
The two acute angles always add up to 90 degrees
Area: Area = (1/2) × a × b
Half the product of the two legs
Practical Examples
Example 1: Finding Trig Values
Find all trigonometric functions for an angle of 30°.
- sin(30°) = 0.5
- cos(30°) ≈ 0.866
- tan(30°) ≈ 0.577
- csc(30°) = 2
- sec(30°) ≈ 1.155
- cot(30°) ≈ 1.732
Example 2: Solving a Right Triangle
A right triangle has one leg a = 3 and hypotenuse c = 5. Find the other side and angles.
- Side b: b² = c² - a² = 25 - 9 = 16, so b = 4
- Angle A: sin(A) = 3/5 = 0.6, so A ≈ 36.87°
- Angle B: B = 90° - 36.87° = 53.13°
- Area: (1/2) × 3 × 4 = 6 square units
Example 3: Finding Building Height
From a point 50 meters from a building, the angle of elevation to the top is 60°. How tall is the building?
- Given: adjacent = 50 m, angle = 60°
- Formula: tan(60°) = height / 50
- Calculation: height = 50 × tan(60°) = 50 × 1.732 ≈ 86.6 m
Trigonometric Identities
Pythagorean Identities:
sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)
Reciprocal Identities:
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)
Quotient Identities:
tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)
Real-World Applications
- Architecture & Construction: Calculating roof slopes, stair angles, and building heights
- Navigation: Determining distances and bearings using angles
- Astronomy: Calculating positions and distances of celestial bodies
- Physics: Analyzing forces, waves, and oscillations
- Engineering: Designing bridges, roads, and mechanical systems
- Computer Graphics: Rendering 3D objects and calculating rotations
- Surveying: Measuring land distances and elevations
Degrees vs Radians
Angles can be measured in two units:
- Degrees (°): A circle has 360 degrees
- Radians (rad): A circle has 2π radians
Conversion formulas:
Radians = Degrees × (π/180)
Degrees = Radians × (180/π)
Frequently Asked Questions
What's the difference between sin, cos, and tan?
They represent different ratios in a right triangle. Sin compares opposite to hypotenuse, cos compares adjacent to hypotenuse, and tan compares opposite to adjacent. Each is useful for different types of problems.
Why is tan(90°) undefined?
Because tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0. Division by zero is undefined in mathematics.
How do I know which trig function to use?
It depends on what you know and what you're trying to find. Use SOH-CAH-TOA: if you have opposite and hypotenuse, use sine; if you have adjacent and hypotenuse, use cosine; if you have opposite and adjacent, use tangent.
What are the reciprocal trig functions used for?
Cosecant (csc), secant (sec), and cotangent (cot) are useful in certain formulas and can simplify calculations. They're less common than sin, cos, and tan but important in advanced mathematics.
Can I use trigonometry with non-right triangles?
Yes! Use the Law of Sines and Law of Cosines for non-right triangles. However, this calculator focuses on right triangles and basic trig functions.
How accurate should my angle measurements be?
For most practical applications, 2 decimal places is sufficient. Engineering and scientific work may require more precision. Always consider the precision of your input measurements.