Math Notes

In the unit circle, we can visualize the relationship between sine and cosine functions. For any angle θ:

• sin(θ) represents the y-coordinate
• cos(θ) represents the x-coordinate
• The fundamental identity sin²(θ) + cos²(θ) = 1 is derived from the Pythagorean theorem

SOH CAH TOA is a mnemonic device to remember the trigonometric ratios for right triangles:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Learn different methods to calculate the area of triangles, including:

• Standard formula: A = ½ × base × height
• Area of right-angled triangles
• Area of equilateral triangles
• Area of isosceles triangles
• Heron's formula for any triangle

Learn how two-dimensional shapes can be rotated around an axis to create three-dimensional objects:

• Understand the concept of solids of rotation
• Explore common shapes like cylinders, cones, and spheres
• Learn volume formulas for various solids
• See how calculus is applied to calculate complex volumes
• Study advanced examples and interactive demonstrations

Master geometric transformations that change size while preserving shape:

• Understand scale factors and centers of dilation
• Calculate area changes with k² scaling relationship
• Determine scale factors from given areas
• Apply dilations to solve real-world problems
• Work with enlargements and reductions

Explore the fundamental relationship between mass, volume, and density:

• Master the density formula: Density = Mass/Volume
• Convert between different units (g/cm³, kg/m³)
• Calculate any variable when given the other two
• Apply density concepts to real-world problems
• Understand density as an intrinsic property

Calculate volumes of various three-dimensional geometric solids:

• Cylinders: V = πr²h or V = B×h
• Cones: V = (1/3)πr²h
• Pyramids: V = (1/3)×BaseArea×h
• Prisms: V = BaseArea×h
• Spheres: V = (4/3)πr³