Question 1

What is the difference between degrees and radians?

Accepted Answer

Degrees and radians are two units for measuring angles. A full circle = 360° = 2π radians. Key conversions: 0° = 0 rad, 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 60° = π/3 ≈ 1.0472 rad, 90° = π/2 ≈ 1.5708 rad, 180° = π ≈ 3.1416 rad. Formula: Radians = Degrees × (π ÷ 180). Degrees = Radians × (180 ÷ π). When to use each: degrees are more intuitive for geometry and everyday angles (right angle = 90°). Radians are used in calculus and physics — the derivative of sin(x) is cos(x) only when x is in radians. Always check your calculator mode before computing trig functions.

Question 2

How do sine, cosine, and tangent work?

Accepted Answer

In a right triangle with angle θ: sin(θ) = opposite ÷ hypotenuse. cos(θ) = adjacent ÷ hypotenuse. tan(θ) = opposite ÷ adjacent = sin(θ) ÷ cos(θ). Memory aid: SOH-CAH-TOA. Key values (memorize): sin(30°) = 0.5, sin(45°) = √2/2 ≈ 0.707, sin(60°) = √3/2 ≈ 0.866. cos(0°) = 1, cos(90°) = 0. tan(45°) = 1, tan(90°) = undefined (vertical asymptote). Inverse functions: arcsin (sin⁻¹), arccos, arctan — these find the angle when you know the ratio. sin⁻¹(0.5) = 30°. The Pythagorean identity: sin²(θ) + cos²(θ) = 1 always.

Question 3

How do logarithms work and what is the natural log?

Accepted Answer

A logarithm answers: "what exponent do I need?" log_b(x) = y means b^y = x. Common logarithm (log or log₁₀): log(100) = 2 because 10² = 100. log(1000) = 3. log(1) = 0. Natural logarithm (ln): uses base e ≈ 2.71828. ln(e) = 1. ln(e²) = 2. ln(1) = 0. ln(2) ≈ 0.693. Key rules: log(a × b) = log(a) + log(b). log(a ÷ b) = log(a) − log(b). log(a^n) = n × log(a). Conversion: ln(x) = log(x) ÷ log(e) = log(x) ÷ 0.4343 = log(x) × 2.3026. Applications: decibels (sound), Richter scale (earthquakes), pH scale (chemistry), compound interest formulas, information theory (bits).

Question 4

How do I calculate exponents and powers?

Accepted Answer

Exponentiation: x^n means x multiplied by itself n times. Examples: 2^10 = 1,024. 3^4 = 81. 10^6 = 1,000,000. Negative exponents: x^(-n) = 1 ÷ x^n. 2^(-3) = 1/8 = 0.125. Fractional exponents: x^(1/2) = √x (square root). x^(1/3) = ∛x (cube root). x^(2/3) = (∛x)². Rules: x^a × x^b = x^(a+b). x^a ÷ x^b = x^(a−b). (x^a)^b = x^(a×b). Scientific notation: 6.022 × 10^23 (Avogadro's number). On a calculator, enter as 6.022 EE 23 or 6.022 × 10^23. Powers of 2 (useful in computing): 2^8 = 256, 2^10 = 1,024 (1 KB), 2^20 = 1,048,576 (1 MB), 2^30 ≈ 1 billion (1 GB).

Question 5

What is a factorial and how is it calculated?

Accepted Answer

Factorial (n!) = n × (n−1) × (n−2) × ... × 2 × 1. By definition: 0! = 1. Examples: 5! = 5 × 4 × 3 × 2 × 1 = 120. 10! = 3,628,800. 20! = 2.43 × 10^18. Factorials grow extremely fast — 100! has 158 digits. Applications: counting permutations (ordered arrangements): n! ways to arrange n items. Combinations: C(n,k) = n! ÷ (k! × (n−k)!) — choosing k items from n without order. 5 choose 2 = 5! ÷ (2! × 3!) = 120 ÷ 12 = 10. Probability: chance of specific card order in a shuffled deck = 1 ÷ 52! ≈ 10^(-68). Stirling's approximation for large n: n! ≈ √(2πn) × (n/e)^n.

Question 6

What are the key mathematical constants (π, e, φ)?

Accepted Answer

π (pi) ≈ 3.14159265358979: ratio of a circle's circumference to its diameter. C = 2πr, A = πr². Appears in trigonometry, probability, Fourier analysis. e (Euler's number) ≈ 2.71828182845905: base of the natural logarithm. Governs continuous exponential growth/decay: A = Pe^(rt). Appears in calculus (derivative of e^x is e^x), compound interest, probability distributions. φ (golden ratio) ≈ 1.61803398875: φ = (1 + √5) ÷ 2. Appears in Fibonacci sequence (consecutive terms approach φ), architecture, nature (leaf spirals, shell growth). √2 ≈ 1.41421356: diagonal of a unit square. √3 ≈ 1.73205081: height of an equilateral triangle with side 2. These constants appear throughout mathematics, physics, and engineering.

Question 7

How do I calculate square roots and nth roots?

Accepted Answer

Square root: √x = x^(1/2). √4 = 2. √9 = 3. √2 ≈ 1.41421. √(a × b) = √a × √b. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cube root: ∛x = x^(1/3). ∛8 = 2. ∛27 = 3. ∛1000 = 10. General nth root: ⁿ√x = x^(1/n). Simplifying radicals: √48 = √(16 × 3) = 4√3. Rationalizing denominators: 1/√2 = √2/2 (multiply numerator and denominator by √2). Distance formula in geometry: d = √((x₂−x₁)² + (y₂−y₁)²) — uses square root. Quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a.

Question 8

What is order of operations (PEMDAS/BODMAS)?

Accepted Answer

PEMDAS (US) / BODMAS (UK) defines the order to evaluate mathematical expressions: Parentheses/Brackets → Exponents/Orders → Multiplication & Division (left to right) → Addition & Subtraction (left to right). Example: 2 + 3 × 4² − (6 ÷ 2). Step 1 (parentheses): 6 ÷ 2 = 3. Step 2 (exponents): 4² = 16. Step 3 (multiplication): 3 × 16 = 48. Step 4 (left to right +/−): 2 + 48 − 3 = 47. Common mistakes: 8 ÷ 2(2+2) = 8 ÷ 2 × 4 = 16 (not 1 — implicit multiplication has the same precedence as explicit multiplication). −3² = −9 (not +9 — the exponent applies before the negation). When in doubt, use parentheses to make the intended order explicit.

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