Trigonometry I Study Notes

5 Trigonometry I Overview

  • Historical context of trigonometry with significant contributions from ancient cultures.

  • Applications in navigation, architecture, astronomy, and more.

Lesson 5.1 Overview

  • Trigonometry means 'triangle measurement'.

  • Used to understand the universe and vital in navigation.

Lesson 5.2 Pythagoras’ Theorem

  • Pythagoras’ theorem states that in a right triangle: a2+b2=c2a^2 + b^2 = c^2 (where c is the hypotenuse).

  • Calculate side lengths with c=rac1extHc = rac{1}{ ext{H}} or a=rac1extOa = rac{1}{ ext{O}}.

Lesson 5.3 Pythagoras’ Theorem in 3D

  • Extends Pythagorean theorem to 3D objects like cuboids and pyramids.

  • Identify right angles for calculation.

Lesson 5.4 Trigonometric Ratios

  • Definitions: Sin, Cos, Tan based on triangle side lengths.

  • Ratios mapped as: extsin(θ)=racOH,extcos(θ)=racAH,exttan(θ)=racOAext{sin}(θ) = rac{O}{H}, ext{cos}(θ) = rac{A}{H}, ext{tan}(θ) = rac{O}{A}.

Lesson 5.5 Using Trigonometry to Calculate Side Lengths

  • Apply ratios to find unknown side lengths with known angle and one side.

Lesson 5.6 Using Trigonometry to Calculate Angle Size

  • Use inverse trigonometric operations to find angles: θ=extsin−1(a),extcos−1(a),exttan−1(a)θ = ext{sin}^{-1}(a), ext{cos}^{-1}(a), ext{tan}^{-1}(a).

Lesson 5.7 Angles of Elevation and Depression

  • Angle of elevation: looking up to an object; angle of depression: looking down to an object.

Lesson 5.8 Bearings

  • Compass bearings indicate direction (N, S, E, W) and true bearings (measured clockwise from north).

  • Example compass bearing: S20°E.

Lesson 5.9 Applications

  • Use trigonometry in practical applications such as predicting bullet trajectories.

Lesson 5.10 Review

  • Summarizes key concepts like Pythagorean theorem, bearings, and trigonometric ratios.

  • Encourages practical application and understanding through problem-solving.