Trig Calculator

Sine and Cosine: The Foundation of Trigonometry

Sine and cosine emerge from the geometry of a right triangle inscribed in a unit circle (a circle with radius 1). When you draw a line from the circle's centre at angle θ to its edge, that line becomes the hypotenuse of a right triangle.

Because the circle has radius 1, the hypotenuse always has length 1. The two legs of the triangle are:

• sin(θ) — the vertical distance from the horizontal axis (the opposite side)
• cos(θ) — the horizontal distance from the vertical axis (the adjacent side)

These ratios hold for any right triangle, not just unit circles. For a triangle with hypotenuse h, opposite side o, and adjacent side a:

• sin(θ) = o ÷ h
• cos(θ) = a ÷ h

Sine and cosine are complementary: sin(θ) = cos(90° − θ), meaning they encode the same information from different perspectives.

Six Trigonometric Functions

Once you know sine and cosine, you can derive the remaining four trig functions using reciprocals and ratios:

• θ (theta) — The angle in question, measured in degrees or radians
• sin(θ) — Sine: the ratio of the opposite side to the hypotenuse
• cos(θ) — Cosine: the ratio of the adjacent side to the hypotenuse
• tan(θ) — Tangent: the ratio of opposite to adjacent
• cot(θ) — Cotangent: the reciprocal of tangent
• sec(θ) — Secant: the reciprocal of cosine
• csc(θ) — Cosecant: the reciprocal of sine

Solving Right Triangles with Trigonometry

When you know one angle and one side length of a right triangle, trigonometry reveals the rest. The key relationships are:

• Finding a side: If you know an angle α and the hypotenuse c, then the opposite side is c × sin(α) and the adjacent side is c × cos(α).
• Finding an angle: If you know two sides, use the inverse functions. For example, α = arctan(opposite ÷ adjacent).
• Complementary angles: In a right triangle, the two non-right angles always sum to 90°. If one angle is α, the other is 90° − α.

The Pythagorean theorem connects all three sides: a² + b² = c², where c is the hypotenuse. You can combine this with trig identities to solve for any missing measurement.

Common Pitfalls and Practical Caveats

Trigonometry is powerful but easy to misapply. Watch for these frequent mistakes:

1. Angle units matter — Calculators and functions evaluate angles differently in degrees vs. radians. π radians = 180°. Always confirm your input units—entering 180 degrees instead of π radians will give wildly incorrect results. Most scientific calculators let you toggle between modes.
2. Tangent and cotangent have discontinuities — tan(θ) is undefined at 90°, 270°, and every odd multiple of 90°. Similarly, cot(θ) is undefined at 0°, 180°, and even multiples of 180°. If your triangle angle approaches these values, tangent or cotangent will blow up to infinity.
3. Check the quadrant for inverse functions — When you use arcsin, arccos, or arctan to find an angle from a ratio, the calculator gives you one answer—but the true angle might be in a different quadrant. In a right triangle this is less of an issue, but in general trigonometry, always verify your angle makes geometric sense.
4. Reciprocals are easy to flip — Remembering which function is which reciprocal causes frequent errors. Secant = 1/cosine and cosecant = 1/sine. Write a quick reference card if you use these less frequently; mixing them up inverts your answer.

The Isosceles Right Triangle (45-45-90)

A 45-45-90 triangle has two equal legs and one right angle. If each leg has length a, then:

• Hypotenuse: c = a√2 (from the Pythagorean theorem: √(a² + a²) = a√2)
• Area: A = a²/2
• Perimeter: P = a(2 + √2) ≈ 3.414a

For any 45-45-90 triangle, sin(45°) = cos(45°) = 1/√2 ≈ 0.707, and tan(45°) = 1. These are among the most useful memorized values in trigonometry.