📘 Math Identities for Pakistan Armed Forces Test.
Mathematical identities are very important for tests like Pak Army, PAF, Navy, Cadet Colleges, and other civil forces. Below are the most important Algebraic and Trigonometric formulas you should memorize.
🧮 A. Basic Algebraic Identities
| Identity Name | Formula |
|---|---|
| Square of Sum | (a + b)² = a² + 2ab + b² |
| Square of Difference | (a − b)² = a² − 2ab + b² |
| Difference of Squares | a² − b² = (a + b)(a − b) |
| Cube of Sum | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Cube of Difference | (a − b)³ = a³ − 3a²b + 3ab² − b³ |
| Product of Sum & Difference | (a + b)(a − b) = a² − b² |
| Expansion of (x + a)(x + b) | x² + (a + b)x + ab |
📏 B. Trigonometric Identities
These are important for solving angle and geometry questions in entry tests.
✅ 1. Pythagorean Identities
| Formula | Description |
|---|---|
| sin²θ + cos²θ = 1 | Basic identity |
| 1 + tan²θ = sec²θ | Used in tangent problems |
| 1 + cot²θ = csc²θ | Used in cotangent problems |
🔁 2. Reciprocal Identities
| Function | Reciprocal |
|---|---|
| sinθ | 1 / cscθ |
| cosθ | 1 / secθ |
| tanθ | 1 / cotθ |
| cscθ | 1 / sinθ |
| secθ | 1 / cosθ |
| cotθ | 1 / tanθ |
➗ 3. Quotient Identities
tanθ = sinθ / cosθ
cotθ = cosθ / sinθ
➕ 4. Sum and Difference Formulas
| Formula Type | Formula |
|---|---|
| sin(A + B) | = sinA·cosB + cosA·sinB |
| sin(A − B) | = sinA·cosB − cosA·sinB |
| cos(A + B) | = cosA·cosB − sinA·sinB |
| cos(A − B) | = cosA·cosB + sinA·sinB |
| tan(A + B) | = (tanA + tanB) / (1 − tanA·tanB) |
| tan(A − B) | = (tanA − tanB) / (1 + tanA·tanB) |
✴️ 5. Double-Angle Formulas
| Function | Formula |
|---|---|
| sin(2A) | = 2·sinA·cosA |
| cos(2A) | = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A |
| tan(2A) | = (2·tanA) / (1 − tan²A) |
⚪ 6. Half-Angle Formulas
| Function | Formula |
|---|---|
| sin²A | = (1 − cos(2A)) / 2 |
| cos²A | = (1 + cos(2A)) / 2 |
| tan²A | = (1 − cos(2A)) / (1 + cos(2A)) |
📌 Pro Tip: Try to memorize with examples and solve past papers. Most MCQs in math portion come from these formulas.
📐 Angles and Triangles for Pakistan Armed Forces Test.
Understanding the types of angles and classifications of triangles is essential for solving geometry questions in Pak Army, Navy, and PAF entrance tests.
🔺 A. Types of Angles
| Angle Type | Description |
|---|---|
| 📏 Acute Angle | Less than 90° |
| ⬛ Right Angle | Exactly 90° |
| 🔻 Obtuse Angle | Greater than 90° but less than 180° |
| ➖ Straight Angle | Exactly 180° |
| 🔄 Reflex Angle | Greater than 180° but less than 360° |
🔁 B. Angle Pair Relationships
| Relationship | Description |
|---|---|
| 🔗 Complementary Angles | Two angles whose sum is 90° |
| 🔗 Supplementary Angles | Two angles whose sum is 180° |
Tip: These pairs are often used in MCQs with right-angle or straight-line diagrams.
🔻 C. Classification of Triangles by Angles
| Triangle Type | Description |
|---|---|
| 🔺 Acute Triangle | All three angles are less than 90° |
| ⬛ Right Triangle | One angle is exactly 90° |
| 🔻 Obtuse Triangle | One angle is more than 90° |
📘 Quick Notes:
In right triangles, the Pythagorean theorem applies.
Triangle angle sum is always 180°.
Use angle classification to deduce side types and solve perimeter/area questions.
🔵 The Circle and Its Parts – Pakistan Armed Forces Math Test
Understanding the key parts of a circle is essential in solving geometry-based MCQs in Pak Army, Navy, and PAF entry tests. Here’s a simple breakdown:
⚪ What is a Circle?
🔘 Circle: A set of all points in a plane that are at an equal distance from a central point called the center.
🧭 Essential Parts of a Circle
| Term | Definition |
|---|---|
| 🧿 Radius | The distance from the center to any point on the circle. |
| 🟡 Diameter | A straight line passing through the center, connecting two points on the circle. It is twice the radius. |
| 🧵 Chord | A line segment connecting two points on the circle without passing through the center. |
| 📏 Tangent | A straight line that touches the circle at only one point. It is perpendicular to the radius at the point of contact. |
| 📐 Secant | A straight line that intersects the circle at two points. |
| 🌀 Arc | A portion of the circle’s circumference. 🔹 Minor Arc: Less than half the circle. 🔸 Major Arc: More than half the circle. |
| 🟠 Sector | A “pizza-slice” shaped region between two radii and the arc. |
| 🟣 Segment | A region between a chord and the arc. |
| 📎 Circumference | The total distance around the circle. 🧮 Formula: C = 2πr or πd |
📘 Quick Formula Review:
Area of Circle: A = πr²
Circumference: C = 2πr
Diameter: D = 2r
🧠 Pro Tip:
In entry test MCQs, you may be asked to identify parts of a circle in diagrams, find area or circumference, or calculate angle sizes in sectors.