Algebra and Geometrical Figures

Study Notes

Algebra, Indices, Number and Surds

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
Indices are used to represent repeated multiplication of a number or variable.
Surds are irrational numbers that cannot be expressed as a simple fraction.

Properties of Geometrical Figures I

Triangles are three-sided polygons with three interior angles.
Quadrilaterals are four-sided polygons with four interior angles.
Circles are two-dimensional shapes where all points are the same distance from the center.

Logarithms and Polynomials

Logarithms are the inverse function of exponentiation.
Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

Properties of Geometrical Figures II

Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Area is the amount of space a two-dimensional shape occupies.
Volume is the amount of space a three-dimensional shape occupies.

Non-Linear Relationships, Functions and their Graphs

Non-linear relationships are relationships between variables that are not represented by a straight line.
Functions are mathematical relationships that map each input value to a unique output value.
Graphs are visual representations of relationships between variables.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Sine, cosine, and tangent are trigonometric functions that relate the angles of a right triangle to the lengths of its sides.
Unit circle is a circle with a radius of 1 used to understand trigonometric functions.

Trigonometric Ratios

Definition: Ratios of the sides of a right triangle relative to its angles
Primary Ratios:
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
Reciprocal Ratios:
- Cosecant (csc): 1/sin = Hypotenuse / Opposite
- Secant (sec): 1/cos = Hypotenuse / Adjacent
- Cotangent (cot): 1/tan = Adjacent / Opposite

Applications Of Trigonometry

Height and Distance:
- Determining the height of an object using angles of elevation or depression.
Angle of Elevation/Depression:
- Uses trigonometric ratios to resolve distances and heights in various scenarios.
Navigation and Geography:
- Calculating angles and distances on maps.
Architecture and Engineering:
- Used in design and construction, where angles and dimensions are critical.

Trigonometric Identities

Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
Reciprocal Identities:
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
Quotient Identities:
- tan(θ) = sin(θ)/cos(θ)
Co-Function Identities:
- sin(90° - θ) = cos(θ)
- tan(90° - θ) = cot(θ)

Graphing Trigonometric Functions

Sine Function (sin):
- Period: 360° (or 2π)
- Range: [-1, 1]
- Starts at (0, 0), peaks at (90°, 1)
Cosine Function (cos):
- Period: 360° (or 2π)
- Range: [-1, 1]
- Starts at (0, 1), crosses zero at (90°, 0)
Tangent Function (tan):
- Period: 180° (or π)
- Range: (-∞, ∞)
- Vertical asymptotes at (90°, n*180°), where n is an integer.

Unit Circle

Definition: A circle with a radius of 1 centered at the origin (0, 0) in the coordinate plane.
Key Angles:
- Degrees: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°
- Radians: 0, π/6, π/4, π/3, π/2, 2π/3, 5π/4, 7π/6, π, 7π/4, 4π/3, 3π/2, 5π/3, 11π/6, 2π
Coordinates:
- Each angle corresponds to coordinates (cos(θ), sin(θ)).
Quadrants:
- 1st: sin, cos > 0
- 2nd: sin > 0, cos < 0
- 3rd: sin, cos < 0
- 4th: sin < 0, cos > 0
Conversions:
- Degrees to Radians: ° × (π/180)
- Radians to Degrees: rad × (180/π)

Description

Test your knowledge on Algebra, Indices, Logarithms, Surds, and various geometrical figures including triangles, quadrilaterals, and circles. This quiz covers fundamental properties and theorems that are essential for mastering mathematics. Engage with concepts that lay the groundwork for advanced mathematical studies.