- Conditional Set $$\{x|x \in \mathbb{N} \land x<10\} = \{1,2,3,4,5,6,7,8,9\}$$
- Union of Sets $$\{1,2,3\} \cup \{2,3,4\} = \{1,2,3,4\}$$
- Intersection of Sets $$\{1,2,3\} \cap \{2,3,4\} = \{2,3\}$$
- Set Complement $$\mathbb{U} = \{1,2,3,4,5,6\}$$ $$\{1,2,3\}' = \{4,5,6\}$$
- Element of $$1 \in \{1,2,3\}$$ $$4 \notin \{1,2,3\}$$
- Subset $$\{1,2,3\} \subset \{1,2,3,4\}$$ $$\{1,3,5\} \not\subset \{1,2,3,4\}$$
- Super set $$\{1,2,3,4\} \supset \{1,2,3\}$$ $$\{1,2,3,4\} \not\supset \{1,3,5\}$$
- Powerset $$P(\{1,2,3\}) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}$$
- Integer $$\mathbb{Z} = \{\cdots,-2,-1,0,1,2,\cdots\}$$
- Positive Integer, Natural Numbers $$\mathbb{Z}^+, \mathbb{N} = \{1,2,3,4,5,\cdots\}$$
- Negative Integer $$\mathbb{Z}^- = \{-1,-2,-3,\cdots\}$$
- Zero Integer $$\mathbb{Z}^0 = \{0\}$$
- Rational Number $$\mathbb{Q} = \{\frac{a}{b} | a,b \in \mathbb{Z}, b \neq 0\}$$ $$\mathbb{Q} = \{0,1,-1,\frac{1}{2},-\frac{1}{2},\frac{1}{3},-\frac{1}{3},\frac{2}{3},-\frac{2}{3},\cdots\}$$
- Irrational Number $$\mathbb{I} = \mathbb{R} \setminus \mathbb{Q} = \{\sqrt{2},\pi,e,\cdots\}$$
- Set Product $$A \times B = \{(a,b) | a \in A, b \in B\}$$ $$\{1,2\} \times \{a,b\} = \{(1,a),(1,b),(2,a),(2,b)\}$$
- Inverse of Relation $$r = \{(x,y) | y = 3x+2\}$$ $$r^{-1} = \{(x,y) | x = \frac{y-2}{3}\}$$
- Domain and Range of Relation $$D(r) = \{x | \exists y, (x,y) \in r\}$$ $$R(r) = \{y | \exists x, (x,y) \in r\}$$ $$r = \{(x,y) | y = x^2\}$$ $$D(r) = \mathbb{R}$$ $$R(r) = \{x|x \in \mathbb{R} \land x > 0\}$$
- Function $$f(x) = x + 1$$ $$f(3) = 4$$
- Inverse of Function $$f(x) = x + 1$$ $$f^{-1}(x) = x - 1$$ $$f^{-1}(4) = 3$$
- Domain and Range of Function $$f(x) = \sqrt{x+3}$$ $$D_f = [-3,\infty)$$ $$R_f = [0,\infty)$$
- Sin - Sine $$\sin(\theta) = \frac{Opposite}{Hypotenuse}$$
- Cos - Cosine $$\cos(\theta) = \frac{Adjacent}{Hypotenuse}$$
- Tan - Tangent $$\tan(\theta) = \frac{Opposite}{Adjacent}$$
- Sec - Secant $$\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{Hypotenuse}{Adjacent}$$
- Csc - Cosecant $$\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{Hypotenuse}{Opposite}$$
- Cot - Cotangent $$\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{Adjacent}{Opposite}$$
- Trigonometry Table
Function $$0^{\circ}$$ $$15^{\circ}*$$ $$30^{\circ}$$ $$45^{\circ}$$ $$60^{\circ}$$ $$75^{\circ}*$$ $$90^{\circ}$$ $$sin(\theta)$$ $$0$$ $$\frac{\sqrt{6}-\sqrt{2}}{4}$$ $$\frac{1}{2}$$ $$\frac{\sqrt{2}}{2}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{\sqrt{6}+\sqrt{2}}{4}$$ $$1$$ $$cos(\theta)$$ $$1$$ $$\frac{\sqrt{6}+\sqrt{2}}{4}$$ $$\frac{\sqrt{3}}{2}$$ $$\frac{\sqrt{2}}{2}$$ $$\frac{1}{2}$$ $$\frac{\sqrt{6}-\sqrt{2}}{4}$$ $$0$$ $$tan(\theta)$$ $$0$$ $$2-\sqrt{3}$$ $$\frac{\sqrt{3}}{3}$$ $$1$$ $$\sqrt{3}$$ $$2+\sqrt{3}$$ $$undefined$$ $$csc(\theta)$$ $$undefined$$ $$\sqrt{6}+\sqrt{2}$$ $$2$$ $$\sqrt{2}$$ $$\frac{2\sqrt{3}}{3}$$ $$\sqrt{6}-\sqrt{2}$$ $$1$$ $$sec(\theta)$$ $$1$$ $$\sqrt{6}-\sqrt{2}$$ $$\frac{2\sqrt{3}}{3}$$ $$\sqrt{2}$$ $$2$$ $$\sqrt{6}+\sqrt{2}$$ $$undefined$$ $$cot(\theta)$$ $$undefined$$ $$2+\sqrt{3}$$ $$\sqrt{3}$$ $$1$$ $$\frac{\sqrt{3}}{3}$$ $$2-\sqrt{3}$$ $$0$$ - Extra Formula
$$\sin^2(\theta) + \cos^2(\theta) = 1$$
$$\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$$ $$\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)$$ $$\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$$ $$\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$$ $$\tan(A+B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$ $$\tan(A-B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$$
$$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$ $$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$ $$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$ $$\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$$ $$\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$$ $$\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos(\theta)}{1+\cos(\theta)}}$$ - Inverse Trigonometry Functions $$y = \sin(x) \quad x = \sin^{-1}(y) = \arcsin(y)$$ $$y = \cos(x) \quad x = \cos^{-1}(y) = \arccos(y)$$ $$y = \tan(x) \quad x = \tan^{-1}(y) = \arctan(y)$$ $$\sin(\sin^{-1}(x)) = x \quad \sin^{-1}(\sin(\theta)) = \theta$$ $$\cos(\cos^{-1}(x)) = x \quad \cos^{-1}(\cos(\theta)) = \theta$$ $$\tan(\tan^{-1}(x)) = x \quad \tan^{-1}(\tan(\theta)) = \theta$$
- Newton's first law of motion - Inertia
Newton's Law of Inertia, states that an object at rest will stay at rest, and an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an external, unbalanced force. This means that a change in an object's state of motion—whether it's moving or stationary—requires a net force to overcome its inertia, which is its resistance to changes in motion. $$\Sigma F_{net} = 0$$ - Newton's second law of motion - Force and Acceleration
Newton's Law of Force and Acceleration states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It is most famously expressed by the equation: $$\Sigma \vec{F}_{ext} = m\vec{a}$$ - Newton's third law of motion - Action and Reaction
Newton's Law of Action and Reaction, states that for every action, there is an equal and opposite reaction. This means that forces always occur in pairs. $$\vec{F}_AB = -\vec{F}_{BA}$$ - Formula to remember $$\Sigma F = m a$$ $$W = mg$$ $$f = \mu N$$ $$f = G\frac{m_1 m_2}{r^2}$$ $$N = mg\cos\theta$$
- In physics, an object is in static equilibrium when it is completely at rest—meaning it has no linear or rotational motion. This state occurs when all the forces and torques acting on the object are perfectly balanced.
- Formula to remember $$\Sigma F_{ext} = 0$$
$$\frac{F_1}{\sin(\alpha)} = \frac{F_2}{\sin(\beta)} = \frac{F_3}{\sin(\gamma)}$$
- Open System
Exchanges both energy and matter with its surroundings.
Example: A pan on a stove with water. Heat enters (energy), and water can evaporate or be added (matter). - Closed System
Definition: Exchanges energy but not matter with its surroundings.
Example: A sealed pressure cooker. It allows heat to be added or removed (energy), but the steam (matter) is contained. - Isolated System
Definition: Exchanges neither energy nor matter with its surroundings.
Example: A perfectly insulated thermos flask. In practice, no system is perfectly isolated, but a thermos approximates one by minimizing heat transfer, keeping the contents warm or cold for a period. - The Formula you need to know
Mass(g) = Moles(mol) * Molar Mass(g/mol)$$Mass(g) = \frac{Moles(mol)}{Molar Mass(g/mol)}$$and Avogadro's Number = $$6.022 \times 10^{23}$$$$Numbers Of Particle = Moles(mol) * Avogadro's Number$$ - Example$$H_2O \rightleftharpoons H_2 + O_2$$$$2H_2O \rightleftharpoons 2H_2 + O_2$$
Balancing equation is to make the number of atoms for each element equal on both the reactant (left) and product (right) sides of the equation
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