Trigonometry 1st Quarter Comprehensive Notes

Definition and Historical Background

• Trigonometry derives from the Greek words “trigonon” (triangle) and “metron” (measure).

• Literal meaning: “measure of triangle.”

• Focus: studies relationships involving lengths and angles in triangles (and by extension, periodic phenomena).

• Historical evolution: traces back to \text{3rd century\,BC}.

  • Hipparchus (c. 190–120 BC) is regarded as the “Father of Trigonometry.”

  • Credited with formalising the science of trigonometric ratios and producing one of the first trigonometric tables.

Ratio and Proportion

• Ratio: comparison of two quantities; expresses relative size.

  • Notation: a:b or \dfrac a b (“a to b” or “a over b”).

• Proportion: an equation stating that two ratios are equal.

  • Notation: a:b::c:d

  • Vocabulary:

  • Antecedent: the first term of a ratio (e.g.
    a in a:b).

  • Consequent: the second term (e.g.
    b in a:b).

  • Means: the two inner terms b and c in a:b::c:d.

  • Extremes: the two outer terms a and d.

Solving a Proportion (generic method)

• Example: 3:10 = x:30

• Steps:

  1. Cross-multiply: 10x = 3 \times 30.

  2. Simplify / isolate x: divide both sides by 10 → x = 9.

  3. Interpretation: when the first quantity is 3 while the second is 10, a comparable pair that keeps the same ratio is 9 to 30.

Application of Proportion – Height & Shadow Problems

• Practical formula (height over shadow length):
  \dfrac{\text{Height of object}}{\text{Shadow length of object}} = \dfrac{\text{Height of reference}}{\text{Shadow length of reference}}

• Given values (tree problem):

  • Tree’s height = unknown h.

  • Tree’s shadow = 2.9\,\text{m}.

  • Reference stick’s height = 2.4\,\text{m}.

  • Reference stick’s shadow = 1.3\,\text{m}.

• Set up proportion: \dfrac h{2.9} = \dfrac{2.4}{1.3}.

• Solution via cross-multiplication:
  1.3h = 2.4 \times 2.9 = 6.96
  h = \dfrac{6.96}{1.3} = 5.4\,\text m.

• Conclusion: The tree is approximately 5.4\,\text{m} tall.

• Significance: Demonstrates real-world power of similar triangles and proportional reasoning (e.g.
  surveying, architecture, cinematography lighting).

Similar Polygons & Triangle Similarity Criteria

• Similar polygons: corresponding angles are equal and corresponding side lengths are proportional.

• Triangle similarity theorems (minimal information to guarantee similarity):

  • SSS (Side–Side–Side): three corresponding side ratios are equal.

  • SAS (Side–Angle–Side): two side ratios are equal and the included angle is congruent.

  • AA (Angle–Angle): two pairs of corresponding angles are congruent (third follows automatically).

• Included angle: the angle formed by two given sides.

• Practical note: Similarity underpins scale drawings, map making, and trigonometric ratio definitions.

Pythagorean Theorem & Converse

• Statement (right triangle only):
  a^2 + b^2 = c^2
  where c is the hypotenuse (side opposite the right angle), a and b are legs.

• Standard demonstration example:

  • Given: a = 3, b = 4, c = 5.

  • Verification: 3^2 + 4^2 = 5^2 \Rightarrow 9 + 16 = 25 \Rightarrow 25 = 25 ✓

• Converse: If for a triangle c^2 = a^2 + b^2 holds, then the triangle is right-angled.

  • Example: 15^2 = 9^2 + 12^2 → 225 = 81 + 144 → 225 = 225 → triangle is right.

• Importance:

  • Fundamental bridge uniting geometry & algebra.

  • Cornerstone for distance formula d = \sqrt{(\Delta x)^2 + (\Delta y)^2}; foundation for trigonometric identities (e.g.
    \sin^2\theta + \cos^2\theta = 1).

Types of Right Triangles (by side lengths)

• Right scalene triangle: No two sides equal; angles all distinct; still contains a 90^\circ angle.

• Right isosceles triangle: Two equal legs (a = b); angles 45^\circ,45^\circ,90^\circ; hypotenuse c = a\sqrt{2}.

• Visual–numerical example (from transcript drawing): legs 9\,\text m and 12\,\text m, hypotenuse 15\,\text m constitute a right scalene triangle.

Connections & Broader Relevance

• Similarity criteria (AA, SAS, SSS) and Pythagorean relationships underpin modern trigonometric function definitions \sin, \cos, \tan.

• Applications span:

  • Architecture & construction (accurate heights, non-accessible lengths).

  • Navigation & astronomy (Hipparchus originally compiled star catalogues using trig relationships).

  • Engineering, computer graphics, robotics (distance & angle calculations).

• Ethical dimension: responsible use of measurement (e.g.
  in land surveying or data modelling) demands honesty and transparency.

Quick Reference Formulas & Notes

• Proportion (cross-product): a:b::c:d \iff ad = bc.

Height–shadow relationship (two similar right triangles): \dfrac{H1}{S1} = \dfrac{H2}{S2}

• Pythagorean theorem: a^2 + b^2 = c^2.

• Converse test for right triangle: verify c^2 = a^2 + b^2.

• Leg–leg–hypotenuse pattern for right isosceles: if legs = \ell, then c = \ell\sqrt{2}.