Algebra and Trigonometry Lecture Notes

The Real Number Set: Denoted by \mathbb{R}, this set includes integers (\mathbb{Z} = {0, \pm 1, \pm 2, \dots}), positive integers (\mathbb{Z}^+), and negative integers (\mathbb{Z}^-).
Closure Properties: \mathbb{R} is closed under addition (x + y \in \mathbb{R}) and multiplication (x \cdot y \in \mathbb{R}).
Fundamental Laws:
1. Identity Elements: x + 0 = x and x \cdot 1 = x.
2. Additive Inverse: For every x, there is a unique -x such that x + (-x) = 0.
3. Multiplicative Inverse: For every x \neq 0, there is a unique x^{-1} such that x \cdot x^{-1} = 1.
4. Associative Laws: x + (y + z) = (x + y) + z and x \cdot (y \cdot z) = (x \cdot y) \cdot z.
5. Commutative Laws: x + y = y + x and x \cdot y = y \cdot x.
6. Distributive Law: x \cdot (y + z) = x \cdot y + x \cdot z.
7. Zero Factor Property: x \cdot y = 0 if and only if x = 0 or y = 0.
Subsets of Real Numbers:
- Rational Numbers (\mathbb{Q}): Ratios of two integers (e.g., 2/3, -0.98123, n/1).
- Irrational Numbers: Real numbers that cannot be expressed as ratios of integers (e.g., \sqrt{2}).
Order Properties (Inequalities):
- Trichotomy: Either x < y, x = y, or x > y.
- Transitivity: If x < y and y < z, then x < z.
- Addition: If x < y, then x + z < y + z.
- Multiplication (Positive): If x < y and z > 0, then zx < zy.
- Multiplication (Negative): If x < y and z < 0, then zx > zy.

Linear Equations: Equations in the form ax + b = 0. They are solved by isolating x via inverse operations.
- Equivalent Equations: Equations with the exact same solution sets.
Quadratic Equations: Equations of the form ax^2 + bx + c = 0, where a \neq 0.
- Form x^2 - p = 0: Solutions are x = \pm \sqrt{p} (only real if p \geq 0).
- Form k(x + r)^2 - p = 0: Solutions are x = -r \pm \sqrt{p/k}.
- The Quadratic Formula: For the general form ax^2 + bx + c = 0, use: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
The Discriminant ($\Delta$): \Delta = b^2 - 4ac.
- If \Delta > 0: Two distinct real solutions.
- If \Delta = 0: One real solution (x = -b/2a).
- If \Delta < 0: No real solutions.

Definitions: A polynomial has the form an x^n + a{n-1} x^{n-1} + \dots + a1 x + a0. n is the degree; a_i are coefficients.
Equal Polynomials: Must have identical terms and coefficients.
Division Algorithm: P(x) = K(x)Q(x) + R(x), where the degree of R(x) is less than \text{deg}(K(x)) or R(x) = 0.
Remainder Theorem: The remainder of P(x) divided by x - a is exactly P(a).
Factor Theorem: x - a is a factor of P(x) if and only if P(a) = 0.
Rational Root Theorem: For integer coefficients, any rational root m/k (reduced) must have m as a factor of a0 and k as a factor of an.

Definition: Numbers in the form a + bi, where i = \sqrt{-1} and i^2 = -1.
Operations:
- Addition: (a1 + b1 i) \pm (a2 + b2 i) = (a1 \pm a2) + (b1 \pm b2)i.
- Multiplication: (a1 + b1 i)(a2 + b2 i) = (a1 a2 - b1 b2) + (a1 b2 + a2 b1)i.
- Conjugate: The conjugate of z = a + bi is \bar{z} = a - bi. z \cdot \bar{z} = a^2 + b^2.
- Absolute Value: |z| = \sqrt{a^2 + b^2}.
- Division: Perform by multiplying numerator and denominator by the conjugate of the denominator.

Definition: A correspondence assigning each input x in domain A to exactly one output f(x) in range B.
Composite Functions: f(g(x)) where the output of g becomes the input of f.
Linear Function Graph: y = ax + b is a line. a is the slope; b is the y-intercept.
- If a > 0: Increasing.
- If a < 0: Decreasing.
Quadratic Function Graph (Parabola): y = ax^2 + bx + c. Vertex at x = -b/2a. The line of symmetry is x = -b/2a.
Exponential Function: y = a^x (a > 0). Domain is all real numbers; range is (0, \infty).
Logarithmic Function: y = \log_a x is the inverse of a^y = x. Domain is (0, \infty).
Transformations:
- f(x + c): Horizontal shift (left if c>0, right if c<0).
- f(x) + C: Vertical shift (up if C>0, down if C<0).
- f(ax): Horizontal shrink (a>1) or expansion (0<a<1).
- Af(x): Vertical expansion (A>1) or shrink (0<A<1).

Acute Angle Functions:
- \sin \alpha = y/r
- \cos \alpha = x/r = \sin(90^\circ - \alpha)
- \tan \alpha = y/x = \sin \alpha / \cos \alpha
- \cot \alpha = x/y = \cos \alpha / \sin \alpha
Special Angles ($\sin, \cos$ values):
- 30^\circ: (1/2, \sqrt{3}/2)
- 45^\circ: (1/\sqrt{2}, 1/\sqrt{2})
- 60^\circ: (\sqrt{3}/2, 1/2)
- 90^\circ: (1, 0)
Identities:
- Pythagorean: \sin^2 \alpha + \cos^2 \alpha = 1.
- Double-Angle: \sin 2\alpha = 2\sin \alpha \cos \alpha; \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha.
- Half-Angle: \sin(\alpha/2) = \pm \sqrt{(1 - \cos \alpha)/2}; \cos(\alpha/2) = \pm \sqrt{(1 + \cos \alpha)/2}.
Laws for General Triangles:
- Law of Sines: \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} = 2r.
- Law of Cosines: a^2 = b^2 + c^2 - 2bc \cos \alpha.
Radians: Measurement based on arc length. n^\circ = (\pi/180) \cdot n radians. Full circle is 2\pi radians.

Distance Formula: d = \sqrt{(x1 - x2)^2 + (y1 - y2)^2}.
Midpoint: M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right).
Line Equations:
- General: ax + by + c = 0.
- Slope-Intercept: y = mx + d.
- Point-Slope: y - y1 = m(x - x1).
Parallel/Perpendicular Lines: Lines are parallel if m1 = m2. They are perpendicular if m2 = -1/m1.
Circle Equation: (x - m)^2 + (y - n)^2 = r^2, where (m, n) is the center and r is the radius.