Pythagoras, Angles and Intro to Trigonometry

Pythagoras’ Theorem (Year-8 Revision)

• Applies only to right-angled triangles.

• Statement (standard classroom wording)

  • $c^2 = a^2 + b^2$ where $c$ is the hypotenuse (longest side), $a,b$ are the shorter sides.

  • Many teachers swap the order, but the hypotenuse must be isolated.

• Identifying the hypotenuse

  • Opposite the $90^{\circ}$ symbol; the right-angle’s ‘arrow’ points to it.

  • In diagrams, the two legs touch the right angle; the hypotenuse never does.

• Side-naming convention used in lesson

  • a < b < c (though swapping $a$ and $b$ is algebraically harmless).

Worked Example 1 – Missing Hypotenuse

• Triangle with legs $5\,\text{cm}$ and $12\,\text{cm}$.

• Hypotenuse is unknown ($x$).
  $x^2 = 5^2 + 12^2 = 25 + 144 = 169$
  $x = \sqrt{169} = 13\,\text{cm}$ (no $\pm$ sign because length cannot be negative).

• Reasonableness check
  12 < 13 < 17 (13 lies between the longer leg and the straight-line sum of legs).

Worked Example 2 – Missing Leg

• Legs: $7\,\text{mm}$ (known), $x$ (unknown).

• Hypotenuse: $13\,\text{mm}$.
  $13^2 = 7^2 + x^2 \implies x^2 = 169 - 49 = 120$
  $x = \sqrt{120} \approx 10.95\,\text{mm}$ (rounded to two dp).
  Exact form: $\sqrt{120}\,\text{mm}$.

Pythagorean Converse (Classification Test)

• Given sides $6,7,9$ – is the triangle right-angled?
  Order them $a=6,\,b=7,\,c=9$ and test $c^2 \stackrel{?}{=} a^2 + b^2$.
  $81 \neq 85$ ⇒ statement is false ⇒ not a right-angled triangle.

• Practical use: validates constructions without needing precise drawing.

Sign Choice When Square-Rooting

• Algebraically, $x^2 = 25$ yields $x=\pm5$.

• In geometry length problems, negative lengths are discarded – context overrides pure algebra.

Angle Measurement Refresher

• Unit: degrees ($^{\circ}$). Right angle = $90^{\circ}$, straight angle = $180^{\circ}$.

• Sub-units for precision

  • $1^{\circ} = 60'$ (minutes).

  • $1' = 60''$ (seconds).

• Analogy: hours : minutes : seconds in time.

  • $0.5^{\circ} = 30'$ just like $0.5\,\text{h} = 30\,\text{min}$.

• Use-cases: navigation, astronomy, surveying – tiny angular errors propagate into huge positional errors.

Calculator Skills (DMS ↔ Decimal)

• Dedicated key labelled $^{\circ}\,'\,"$, DMS, or ENG-shift on Casio.

• Conversion decimal → DMS

  1. Type value, press DMS, then $=.$
     Example: $32.8^{\circ}$ → $32^{\circ}48'$.

• Conversion DMS → decimal

  1. Type $82^{\circ}$ DMS $54'$ DMS, press DMS again → $82.9^{\circ}$.

• Watch out: for angles < $1^{\circ}$ you must type the leading $0^{\circ}$, e.g. $0^{\circ}55'12''$.

• Ensure calculator mode is DEG (not RAD or GRAD). Look for “DEG” or “D” indicator; clear/setup otherwise.

Naming Sides Relative to an Angle

• Given a reference angle $\theta$ (usually Greek theta):

  • Hypotenuse (hyp) – longest side, opposite the right angle.

  • Adjacent (adj) – side that touches $\theta$ (excluding the hypotenuse).

  • Opposite (opp) – side across from $\theta$.

• Abbreviations used in notes: hyp / adj / opp or H / A / O.

Trigonometric Ratios (SOH-CAH-TOA)

• “Trigonometry” = triangle-measurement (Greek roots trigonon + metron).

• Three primary ratios (to be memorised!)

  • $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$

  • $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$

  • $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

• Mnemonic: SOH-CAH-TOA
  S O H C A H T O A
  (first letter = function, next two = numerator/denominator order).

• Alternative memory aids: craft sentences from the nine letters, e.g. “Some Old Horse Caught Another Horse Taking Oats Away”.

Calculator Note

• $\sin,\cos,\tan$ keys assume input in degrees unless mode switched.

• Later lessons will cover inverse functions ($\sin^{-1}, \cos^{-1}, \tan^{-1}$) to find angle size from a ratio.

Worked Trigonometric Ratio Examples

Example A – All Three Ratios

Triangle: $\text{opp}=12$, $\text{adj}=16$, $\text{hyp}=20$.

• $\sin\theta = \dfrac{12}{20}=\dfrac{3}{5}$

• $\cos\theta = \dfrac{16}{20}=\dfrac{4}{5}$

• $\tan\theta = \dfrac{12}{16}=\dfrac{3}{4}$
  Future use: feed any ratio into $\sin^{-1}, \cos^{-1}, \tan^{-1}$ to obtain $\theta$ in degrees-minutes-seconds.

Example B – Ratio Requiring Pythagoras First

• Known: $\text{opp}=8$, $\text{adj}=6$, $\text{hyp}=\,?$
  $c^2=6^2+8^2 \Rightarrow c=10.$

• Task: $\sin\theta$.
  $\sin\theta = \dfrac{8}{10}=\dfrac{4}{5}.$

• Demonstrates interplay between Pythagoras and trig.

Scale Factor & Angle Invariance

• Enlarging/shrinking a triangle multiplies all side lengths by the same scale factor but leaves every angle unchanged.

  • e.g. $3!:!4!:!5$ triangle doubled to $6!:!8!:!10$ still has identical angle set.

• Logical proof: Sum of angles must stay $180^{\circ}$; scaling sides cannot change that.

• Significance: trig ratios depend only on angle – the specific size of a similar triangle is irrelevant.

Connections, Implications & Good Practice

• Real-world links

  • Navigation: tiny angular errors over ocean ⇒ kilometre-scale positional drift.

  • Astronomy: stellar positions given in DMS; telescopes rely on precise trigonometry.

• Ethical/Professional expectations

  • Accurate unit/mode use prevents engineering failures (bridges, surveying, aviation approach paths).

  • Always show formula → substitution → evaluation to make work auditable.

• Philosophical reflection

  • Geometry bridges the abstract (numbers) with the tangible (lengths, angles), illustrating mathematics’ descriptive power.

Key Formulae & Study Checklist

• Memorise:
  $c^2=a^2+b^2$ (right-angled only)
  $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$
  $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$
  $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

• Be fluent converting between degrees-decimal and DMS.

• Practise identifying hyp-adj-opp relative to any marked angle.

• Keep calculator in DEG mode; verify before every trig computation.

• Anticipate next lessons: using inverse trig to solve for unknown angles, and applying trig in contextual problems (height/ distance, bearings, slope).