Mathematics AI: Geometry and Trigonometry Study Notes
Hello future mathematical explorers! Welcome to the section of Mathematics Applications and Interpretation (AI) that deals with shape, space, and measurement: Geometry and Trigonometry.
Why is this chapter important? Because geometry is how we build bridges, calculate distances on maps, design products, and navigate the globe. In AI, the focus is heavily on applying these tools to real-world problems—from finding the volume of grain in a silo to measuring the height of a skyscraper without ever climbing it.
Don't worry if shapes and formulas sometimes feel abstract. We will break them down using practical examples and simple steps. Grab your calculator—you’ll need it!
1. Measuring Flat Shapes (2D Geometry)
In AI, while you need to know the basics (area of a rectangle, triangle), the focus often shifts to more complex shapes and parts of circles.
1.1 Perimeter and Area Basics
The Perimeter is the total distance around the outside of a shape. The Area is the measure of the space enclosed within the shape.
- Triangle Area: \(A = \frac{1}{2} \times \text{base} \times \text{height}\)
- Trapezoid (Trapezium) Area: \(A = \frac{1}{2} (a + b) h\), where \(a\) and \(b\) are the parallel sides.
- Circle Area: \(A = \pi r^2\)
- Circle Circumference: \(C = 2\pi r\)
1.2 Circular Measurement: Arcs and Sectors
When dealing with circles, you often need to find the length of a curved edge (arc) or the area of a slice (sector).
Key Concept: Radians vs. Degrees
Mathematics often uses radians, where \(360^{\circ} = 2\pi\) radians. It is crucial to check your calculator mode and use the appropriate formula provided in your formula booklet.
Arc Length (L): The length of the crust of the pizza slice.
If the angle \(\theta\) is in degrees:
$$L = \frac{\theta}{360} \times 2\pi r$$
If the angle \(\theta\) is in radians:
$$L = r\theta$$
Area of a Sector (A): The area of the entire pizza slice.
If the angle \(\theta\) is in degrees:
$$A = \frac{\theta}{360} \times \pi r^2$$
If the angle \(\theta\) is in radians:
$$A = \frac{1}{2} r^2 \theta$$
Quick Review Tip: Always assume you should use the radian formulas (\(L = r\theta\) and \(A = \frac{1}{2} r^2 \theta\)) unless the problem specifically gives the angle in degrees or you are asked to convert. Your calculator can usually handle the conversion for you.
Key Takeaway: Be mindful of units (cm, m, km) and whether your angle input needs to be in degrees or radians. This is a common source of error!
2. Working in Three Dimensions (3D Geometry)
AI heavily involves calculating the volume and surface area of 3D objects, especially those found in engineering and architecture.
2.1 Volume (V) and Surface Area (SA)
Volume measures the space contained inside the 3D object (e.g., how much water fits in a tank). Surface Area measures the total area covering the outside of the object (e.g., how much paint is needed).
You must be comfortable with the following shapes, as their formulas are provided in your booklet:
- Prism/Cylinder: \(V = \text{Area of Base} \times \text{height}\)
- Pyramid/Cone: \(V = \frac{1}{3} \times \text{Area of Base} \times \text{height}\)
- Sphere: \(V = \frac{4}{3} \pi r^3\) and \(SA = 4\pi r^2\)
2.2 Dealing with Composite Solids
Many real-world problems involve composite solids—shapes made up of two or more simple shapes (like an ice cream cone, which is a cone and a hemisphere).
Step-by-Step for Composite Solids:
- Volume: Calculate the volume of each component shape separately, then add them together.
- Surface Area: This is trickier! Calculate the area of only the exposed surfaces. You must NOT include the area where the two shapes join (the shared internal base).
Example: If a hemisphere sits on top of a cylinder, the circular base of the hemisphere and the top circular face of the cylinder are covered and must be subtracted from the total surface area calculation.
Common Mistake to Avoid: When finding the surface area of a composite solid like a hemisphere and a cone joined together, students often forget that the "base" of the cone and the "base" of the hemisphere are now internal surfaces and should be ignored.
Key Takeaway: Volume is always additive. Surface Area requires careful visualization to ensure you only count the parts that touch the air or outside environment.
3. Coordinate Geometry: Mapping the World
Coordinate geometry allows us to solve geometric problems using algebra. Think of it as GPS for math problems!
For two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\), we can find several crucial relationships:
3.1 Distance and Midpoint
Distance between two points: This uses the Pythagorean theorem to find the length of the line segment.
$$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$Midpoint of the segment: The point exactly halfway between A and B.
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$Memory Aid: Midpoint is the average of the coordinates, while distance involves square roots and subtraction (like the Pythagorean theorem).
3.2 Gradient (Slope) and Linear Equations
The gradient (\(m\)) tells us the steepness and direction of a line.
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$Parallel and Perpendicular Lines
- Parallel Lines: They have the same gradient. \(m_1 = m_2\).
- Perpendicular Lines: The product of their gradients is \(-1\). \(m_1 \times m_2 = -1\). (If one gradient is \(m\), the perpendicular gradient is \(-\frac{1}{m}\)).
Equation of a Straight Line
The equation is usually expressed as \(y = mx + c\), where \(m\) is the gradient and \(c\) is the y-intercept.
If you have a gradient \(m\) and a point \((x_1, y_1)\), you can use the point-gradient form (which is often quicker in AI problems):
$$y - y_1 = m(x - x_1)$$
Did you know? Coordinate geometry is essential in computer programming for graphics and robotics, where every movement and interaction needs precise coordinates.
Key Takeaway: Coordinate geometry is the algebra of shapes. Make sure you can switch seamlessly between the three major formulas: distance, midpoint, and gradient.
4. Trigonometry in Right-Angled Triangles
Trigonometry relates angles to side lengths. For right-angled triangles, we use the classic relationships:
4.1 SOH CAH TOA
Remember this mnemonic device:
- SOH: \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- CAH: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
- TOA: \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)
Remember: The Hypotenuse is always the longest side, opposite the right angle. Opposite and Adjacent sides depend on which angle \(\theta\) you are focused on.
4.2 Angles of Elevation and Depression
These terms are critical in application problems involving architecture, surveying, or navigation.
- Angle of Elevation: The angle measured upwards from the horizontal line of sight to an object. (You are looking up.)
- Angle of Depression: The angle measured downwards from the horizontal line of sight to an object. (You are looking down.)
Important Trick for Depression Angles: The angle of depression from point A to point B is equal to the angle of elevation from point B to point A (due to the "Z" rule for parallel lines and transversals). Always draw the horizontal line clearly!
Key Takeaway: Always identify the angle, the side you know, and the side you need to find. This determines whether you use Sine, Cosine, or Tangent.
5. Trigonometry in Non-Right-Angled Triangles
When you don't have a right angle, SOH CAH TOA won't work. You must use the Sine Rule or the Cosine Rule.
In a triangle ABC, side \(a\) is opposite angle A, side \(b\) is opposite angle B, and so on.
5.1 The Sine Rule
The Sine Rule is used when you have a matching pair of known side and angle (e.g., side \(a\) and angle A). It is used in ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or SSA (Side-Side-Angle) situations.
To find a side:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$To find an angle:
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$Common Mistake (HL focus): The Ambiguous Case (SSA) If you are given two sides and a non-included angle (SSA), there might be two possible triangles. If you find an acute angle \(\theta\), the supplementary angle \((180^{\circ} - \theta)\) might also be a solution. You must check if the second angle is geometrically possible within the triangle's angle sum.
5.2 The Cosine Rule
The Cosine Rule is used when you do NOT have a matching side-angle pair. It works for SAS (Side-Angle-Side) or SSS (Side-Side-Side) situations.
To find an unknown side (SAS):
$$a^2 = b^2 + c^2 - 2bc \cos A$$To find an unknown angle (SSS): You need to rearrange the formula. This is often the most useful version in applications:
$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$Analogy: Think of the Cosine Rule as the Pythagorean theorem with a correction factor (\(- 2bc \cos A\)) to account for the angle not being \(90^{\circ}\). If \(A=90^{\circ}\), \(\cos A = 0\), and the formula simplifies back to \(a^2 = b^2 + c^2\).
5.3 Area of a Triangle (Non-Right-Angled)
If you know two sides and the angle included between them (SAS), you can find the area without the height:
$$A = \frac{1}{2} ab \sin C$$Quick Review Box: Which Rule to Use?
- Right Angle? SOH CAH TOA (or Pythagoras).
- Matching Side/Angle Pair? Sine Rule.
- SAS or SSS (No matching pair)? Cosine Rule.
Key Takeaway: Applied trigonometry requires visualizing the scene (often in 3D, breaking it into 2D triangles), correctly identifying the given information, and selecting the appropriate rule (Sine, Cosine, or Area).
6. Geometry and Trigonometry in 3D Space
Many AI problems combine 3D shapes with trigonometry. You are asked to find distances or angles within solids (like pyramids, cubes, or rooms).
Step-by-Step for 3D Problems:
- Visualize and Isolate: Identify the relevant triangle hidden inside the 3D shape.
- The Floor is Your Friend: Most often, the 2D base or "floor" of the shape contains one of the necessary side lengths for your 3D triangle. Use Pythagoras' theorem in the base first to find that length (e.g., the diagonal of the floor).
- Apply Trig: Use SOH CAH TOA, Sine Rule, or Cosine Rule in the 3D triangle you have isolated.
Example: Finding the angle a pyramid's edge makes with its base. You first need the diagonal distance across the base (Pythagoras), and the height of the pyramid. These three lines form a right-angled triangle.
Encouraging Word: Don't worry if this seems tricky at first. Drawing large, clear diagrams and sketching the hidden triangles separately is the most important skill here. Practice making the jump from 3D object to 2D triangle!