2.3 Geometry & Trigonometry

Introduction to Geometry and Trigonometry

Geometry deals with the properties and relations of points, lines, surfaces, and solids. It helps us map out the spatial relationships of the physical world. Trigonometry focuses specifically on the relationships between the side lengths and angles of triangles, particularly right triangles. Both disciplines are heavily tested on the exam.

Angles and Lines

Understanding fundamental angle relationships is the first step in geometry.

• Supplementary angles: Two angles that add up to 180°. Together, they form a straight line.
• Complementary angles: Two angles that add up to 90°. Together, they form a right angle.
• Vertical angles: Opposite angles formed by the intersection of two straight lines. Vertical angles are always equal in measure.
• Parallel lines and transversals: When a line (called a transversal) intersects two parallel lines, several important angle pairs are formed. Corresponding angles are equal, alternate interior angles are equal, and alternate exterior angles are equal.

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Triangles

The golden rule of triangles is that the sum of the interior angles of any triangle is always 180°.

Types of Triangles

• Equilateral: All three sides are equal, and all three angles are equal (60° each).
• Isosceles: Two sides are equal, and the two angles opposite those sides are also equal.
• Scalene: No sides are equal, and no angles are equal.
• Right Triangle: Contains exactly one 90° angle. The side opposite the 90° angle is the longest side and is called the hypotenuse.

The Pythagorean Theorem

For any right triangle with legs $a$ and $b$, and hypotenuse $c$, the theorem states:

$$a^2 + b^2 = c^2$$

Common Pythagorean triples include 3-4-5, 5-12-13, and 8-15-17 (and their multiples, like 6-8-10). Memorizing these triples can save you valuable time on the exam.

Special Right Triangles

There are two special right triangles with unique side ratio properties:

• 45°-45°-90° Triangle: An isosceles right triangle. The sides are in the ratio $x : x : x\sqrt{2}$.
• 30°-60°-90° Triangle: Formed by cutting an equilateral triangle in half. The sides are in the ratio $x : x\sqrt{3} : 2x$, where $x$ is the side opposite the 30° angle, $x\sqrt{3}$ is opposite the 60° angle, and $2x$ is the hypotenuse.

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Polygons and Area

Polygons

A polygon is a closed figure with straight sides. The sum of the interior angles of an $n$-sided polygon is given by the formula: $(n - 2) \times 180°$. For a regular polygon (where all sides and angles are equal), each individual interior angle is $\frac{(n - 2) \times 180°}{n}$.

Area Formulas to Memorize

• Rectangle: $A = l \times w$ (length × width)
• Triangle: $A = \frac{1}{2} b h$ (half base × height)
• Parallelogram: $A = b h$ (base × vertical height)
• Trapezoid: $A = \frac{1}{2} (b_1 + b_2) h$ (average of the parallel bases × height)

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Circles

Key terminology for circles:

• Radius ($r$): The distance from the center point to any point on the edge.
• Diameter ($d$): The distance across the circle passing directly through the center. $d = 2r$.
• Circumference ($C$): The total distance around the outside of the circle. $C = 2\pi r$ or $C = \pi d$.
• Area ($A$): The space inside the circle. $A = \pi r^2$.

An arc is a portion of the circumference. A sector is a portion of the area (like a slice of pizza). To find the length of an arc or the area of a sector, figure out what fraction of the full 360° circle you are dealing with, and multiply that fraction by the total circumference or total area.

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Solid Geometry (Volume)

Volume measures the 3D space an object occupies.

• Rectangular Solid (Box): $V = l \times w \times h$
• Cube: $V = s^3$ (where $s$ is the side length)
• Cylinder: $V = \pi r^2 h$ (Area of the circular base × height)

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Trigonometry

Basic trigonometric ratios apply to right triangles. The mnemonic SOH CAH TOA is crucial for remembering the definitions:

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

Example: In a right triangle, if the angle $\theta$ has an opposite side of length 3 and an adjacent side of length 4, find $\sin(\theta)$.

1. First, find the length of the hypotenuse using the Pythagorean theorem: $3^2 + 4^2 = 9 + 16 = 25$, so the hypotenuse is $\sqrt{25} = 5$.
2. Use the SOH definition: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}$.

Common Traps to Avoid

• Trap: Forgetting to halve the diameter to find the radius when calculating the area of a circle. Always double-check if the problem gives you the radius or the diameter.
• Trap: Confusing the formulas for circumference ($2\pi r$) and area ($\pi r^2$).
• Trap: Trying to apply the Pythagorean theorem or SOH CAH TOA to a triangle that is not a right triangle.