South 고등학교 2학년 Algebra
Chapters: 3
1. Exponential and logarithmic functions
Powers and roots · Rational and real exponents · Exponent laws · Logarithms and their properties · Common logarithms in real life · Exponential and log functions · Graphs of exponential and log functions · Solving problems with exponentials and logs
- Laws of Exponents – An exponent tells how many times a base is multiplied by itself. Same base: multiply → add exponents, divide → subtract, power of a power → multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) is the nth root.
- Logarithms – A logarithm tells you the power you need. log_b x = y means b^y = x. So log₂ 8 = 3 because 2³ = 8. Logs turn multiplying into adding: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a. Base 10 is the common log, base e is the natural log (ln). Any base can be changed: log_b x = log x ÷ log b.
- Exponential Functions – An exponential function has the form y = a·bˣ, where a ≠ 0 is the starting value and b > 0, b ≠ 1 is the growth factor. If b > 1 it grows; if 0 < b < 1 it decays. Each step of 1 in x multiplies y by b (a constant ratio), unlike a linear function which adds a constant. The graph of y = bˣ passes through (0, 1), has domain all real numbers, range y > 0, and the x-axis as a horizontal asymptote. The general form y = a·b^(k(x − d)) + c stretches, reflects and shifts it.
2. Trigonometric functions
General angles and radians · Sine, cosine, tangent functions and graphs · Sine and cosine rules
- Trigonometric Functions: Radians, Unit Circle, Graphs and Identities – An angle of one radian cuts an arc equal to the radius, so π radians = 180°. On a unit circle, the point at angle x is P = (cos x, sin x), which gives sin²x + cos²x = 1 and extends sine and cosine to every real number. The signs follow 'All, Sin, Tan, Cos' in quadrants I to IV; sin x and cos x repeat every 2π and stay between −1 and 1. Compound-angle formulas such as cos(A + B) = cos A cos B − sin A sin B lead to tan(A + B), cot(A + B), sum-to-product, double-angle and triple-angle identities.
- Law of Sines (Sine Rule) – In any triangle, each side divided by the sine of the angle facing it gives the same number: a/sinA = b/sinB = c/sinC = 2R, where R is the radius of the circle through the three corners. Use it when you know two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). SSA can give two triangles, one or none.
3. Sequences
What a sequence is · Arithmetic sequences and sums · Geometric sequences and sums · Sigma notation · Sums of various sequences · Recursive definitions · Mathematical induction
- Sequences and Series: AP, GP, AM and GM – A sequence is a list of numbers in a definite order: a₁, a₂, a₃, … A series is what we get when we add the terms. In an arithmetic progression (AP) we add the same number d each time; aₙ = a + (n − 1)d and Sₙ = n/2[2a + (n − 1)d]. The arithmetic mean (AM) of a and b is (a + b)/2; n AMs between a and b use d = (b − a)/(n + 1). In a geometric progression (GP) we multiply by the same number r each time; aₙ = arⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1. If |r| < 1 the terms shrink and the infinite sum is S∞ = a/(1 − r). The geometric mean (GM) of two positive numbers a and b is √(ab), and n GMs between them use r = (b/a)^(1/(n+1)). For positive a and b, AM ≥ GM, with equality only when a = b.
- Sigma Notation (∑) – The sign ∑ is a short way to write "add up these terms". ∑ from k = 1 to n of a_k means a_1 + a_2 + ... + a_n. Constants come out of the sum, sums can be split, and there are ready formulas for the sums of k, k² and k³. Telescoping lets most terms cancel.
- Recurrence Relations – A recurrence relation makes each term of a sequence from the term (or terms) before it, for example u(n+1) = u(n) + 3 with u(0) = 2. You always need a rule and a starting value. Rules like u(n+1) = u(n) + d give arithmetic sequences, u(n+1) = r·u(n) give geometric ones, and u(n+1) = a·u(n) + b settles at the fixed point b ÷ (1 − a) when −1 < a < 1.
- Proof by Mathematical Induction – Mathematical induction proves that a statement P(n) is true for every natural number n. Step 1 (base case): show P(1) is true. Step 2 (inductive step): assume P(k) is true for some k, and use it to show P(k + 1) is true. Then, like a line of dominoes, P(1) makes P(2) true, P(2) makes P(3) true, and so on for ever.