Trigonometric functions (sin, cos, tan, etc.) relate the angles of a right‑angled triangle to the ratios of its sides. Their inverse functions return the angle for a given ratio. Fundamental trigonometric identities, such as sin²θ + cos²θ = 1, are true for all angles and are essential tools in calculus, geometry, and physics.
Theory & Key Formulas
For an angle θ, the six trigonometric ratios are defined on the unit circle. The most fundamental identity is the Pythagorean identity:
sin²θ + cos²θ = 1
From this, dividing by cos²θ gives sec²θ − tan²θ = 1, and dividing by sin²θ gives cosec²θ − cot²θ = 1. The reciprocal identities are:
secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ
Inverse trigonometric functions are restricted to principal value ranges: arcsin in [−90°, 90°], arccos in [0°, 180°], arctan in (−90°, 90°).
Solved Examples
Example 1: Trig Values for 30°
sin(30°) = 0.5
cos(30°) = √3/2 ≈ 0.866025
tan(30°) = 1/√3 ≈ 0.577350
Example 2: Inverse Trig – arcsin(0.5)
arcsin(0.5) = 30° (principal value)
Example 3: Identity Check at 45°
sin(45°) = cos(45°) ≈ 0.707107
sin² + cos² = 0.5 + 0.5 = 1 ✓
Frequently Asked Questions
Q1. What units can I use for angles?
Degrees and radians. The calculator converts internally; select the appropriate unit from the dropdown. Use decimal numbers like 30 or 0.5236.
Q2. What happens if I enter an out‑of‑range value for inverse trig?
arcsin and arccos require values in [−1, 1]. The calculator will show an error if you enter a number outside this range. arctan accepts any real number.
Q3. Does the identity verifier check for all identities?
It checks the four fundamental Pythagorean‑based identities: sin²+cos²=1, tan=sin/cos, sec²−tan²=1, and cosec²−cot²=1. If any fail, it means the angle is not in the domain of those functions.
Q4. How precise are the results?
Results are rounded to 6 decimal places, using JavaScript's double‑precision (about 15 significant digits). This is sufficient for all JEE and NCERT problems.
Q5. Can I enter angles like 90° or 270°?
Yes, but tan(90°) and sec(90°) are undefined (approach infinity). The calculator will display "Undefined" for those values.