Geometry and Trigonometry

Subject: Mathematics
Topic: 3
Cambridge Code: 0580

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Angles and Lines

Angle Relationships

Angles on straight line: Sum to 180° Angles around point: Sum to 360° Vertically opposite angles: Equal

Parallel Lines and Transversals

When two parallel lines cut by transversal:

Corresponding angles: Equal Alternate angles: Equal Co-interior (allied) angles: Sum to 180°

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Triangles

Types and Properties

Equilateral:

• All sides equal
• All angles 60°
• Height = $\frac{\sqrt{3}}{2}$ × side

Isosceles:

• Two equal sides
• Two equal base angles
• Height bisects base

Right-angled:

• One angle 90°
• Pythagorean theorem applies
• $a^2 + b^2 = c^2$

Triangle Angles

Angle sum: Always 180°

Exterior angle: Equals sum of non-adjacent interior angles

Congruence and Similarity

Congruent triangles: Identical (same size and shape)

• SSS: All three sides equal
• SAS: Two sides and included angle equal
• ASA: Two angles and included side equal
• RHS: Right angle, hypotenuse, and side equal

Similar triangles: Same shape, different size

• AAA (or AA): Angles equal
• SSS: Sides proportional
• SAS: Two sides proportional and included angle equal

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Trigonometric Ratios

Right-Angled Triangle

For angle θ:

$\sin θ = \frac{\text{opposite}}{\text{hypotenuse}}$

$\cos θ = \frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan θ = \frac{\text{opposite}}{\text{adjacent}}$

Memory aid: SOHCAHTOA

Trigonometric Values

Special angles:

Angle	0°	30°	45°	60°	90°
sin	0	1/2	√2/2	√3/2	1
cos	1	√3/2	√2/2	1/2	0
tan	0	1/√3	1	√3	∞

Reciprocal Ratios

$\csc θ = \frac{1}{\sin θ}$

$\sec θ = \frac{1}{\cos θ}$

$\cot θ = \frac{1}{\tan θ}$

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Non-Right Triangles

Sine Rule

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Use when:

• Knowing two angles and one side (ASA/AAS)
• Knowing two sides and non-included angle (SSA) - may give two solutions

Cosine Rule

$a^2 = b^2 + c^2 - 2bc\cos A$

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Use when:

• Knowing two sides and included angle (SAS)
• Knowing all three sides (SSS)

Area of Triangle

$\text{Area} = \frac{1}{2}ab\sin C$

Where a and b are sides and C is included angle.

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Circles

Properties

Circumference: $C = 2πr = πd$

Area: $A = πr^2$

Arc length: $l = \frac{θ}{360°} × 2πr$

Sector area: $A = \frac{θ}{360°} × πr^2$

Circle Theorems

Angle in semicircle: 90°

Angles subtended by same arc: Equal (at circumference)

Angle at center: Twice angle at circumference

Opposite angles in cyclic quadrilateral: Sum to 180°

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Polygons

Interior Angles

Sum of interior angles: $(n - 2) × 180°$ where n = number of sides

Each interior angle (regular): $\frac{(n-2) × 180°}{n}$

Exterior Angles

Sum of exterior angles: Always 360°

Each exterior angle (regular): $\frac{360°}{n}$

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Areas and Perimeters

2D Shapes

Shape	Area	Perimeter
Rectangle	$l × w$	$2(l + w)$
Triangle	$\frac{1}{2}bh$	Sum of sides
Circle	$πr^2$	$2πr$
Trapezoid	$\frac{1}{2}(a+b)h$	Sum of sides
Parallelogram	$bh$	Sum of sides

3D Shapes and Volumes

Shape	Volume	Surface Area
Rectangular prism	$l × w × h$	$2(lw + lh + wh)$
Sphere	$\frac{4}{3}πr^3$	$4πr^2$
Cylinder	$πr^2h$	$2πr^2 + 2πrh$
Cone	$\frac{1}{3}πr^2h$	$πr^2 + πrl$
Pyramid	$\frac{1}{3} × \text{base} × h$	Base + triangular faces

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3D Geometry

Coordinate System

3D coordinates: $(x, y, z)$

Distance between points: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

Lines and Planes

Lines: Defined by point and direction

Planes: Defined by three points or normal vector

Angles between lines/planes: Use vectors and trigonometry

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Trigonometric Functions

Sine and Cosine Curves

$y = \sin x$:

• Period: 360° (2π rad)
• Range: [-1, 1]
• Passes through origin

$y = \cos x$:

• Period: 360°
• Range: [-1, 1]
• Maximum at x = 0°

Tangent Curve

$y = \tan x$:

• Period: 180°
• Range: All real numbers
• Asymptotes at ±90°, ±270°, ...

Transformations

$y = a\sin(bx + c) + d$:

• a: Amplitude
• b: Affects period (period = 360°/b)
• c: Phase shift
• d: Vertical shift

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Key Points

1. Pythagorean theorem for right triangles
2. SOHCAHTOA for trigonometric ratios
3. Sine rule for non-right triangles (ASA, SSA)
4. Cosine rule for non-right triangles (SAS, SSS)
5. Area using $\frac{1}{2}ab\sin C$
6. Circle area and circumference formulas
7. Interior angles sum: $(n-2) × 180°$
8. Circle theorems for angle relationships
9. 3D volume and surface area formulas
10. Trigonometric transformations affect graphs

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Practice Questions

1. Use Pythagorean theorem
2. Find trigonometric ratios
3. Apply sine rule
4. Apply cosine rule
5. Calculate triangle areas
6. Use circle theorems
7. Find areas and perimeters
8. Calculate 3D volumes
9. Solve 3D geometry problems
10. Sketch trigonometric curves

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Revision Tips

• Memorize special angle values
• Understand SOHCAHTOA thoroughly
• Practice sine and cosine rules
• Know when to use each rule
• Understand circle theorems
• Practice 3D visualizations
• Know volume formulas
• Sketch curves accurately
• Use triangle properties