Applications of Trigonometry

Trigonometry is widely used to solve problems involving right-angled and non-right-angled triangles in various real-life contexts. It combines foundational principles like Pythagoras’s theorem, trigonometric ratios, and concepts such as angles of elevation and depression.

Key Concepts

• Right-angled Trigonometry:
  • Application of sine, cosine, and tangent ratios to solve problems involving right-angled triangles.
  • Pythagoras’s theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
• Non-right-angled Trigonometry:
  • Use of the sine rule: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
  • Use of the cosine rule: \( c^2 = a^2 + b^2 – 2ab\cos C \).
  • Area of a triangle: \( \text{Area} = \frac{1}{2}ab\sin C \).
• Angles of Elevation and Depression:
  • Angles measured above (elevation) or below (depression) the horizontal line.
  • Used to solve problems involving heights and distances.
• Bearings:
  • Used in navigation and triangulation, measured clockwise from the north direction.
• Construction of Diagrams:
  • Interpreting and creating labelled diagrams from written statements.

Guidance, Clarifications, and Syllabus Links

• Applications include real-world problems such as triangulation, map-making, navigation, and radio transmissions.
• Includes the use of parallax in navigation.
• Links to physics concepts like vectors, scalars, forces, and dynamics, as well as scientific field studies.
• International-mindedness: Triangulation was historically used to calculate Earth’s curvature and settle disputes like Newton’s theory of gravity.
• Connections to Theory of Knowledge (TOK): How the sum of triangle angles varies depending on the geometry (less than, equal to, or more than \( 180^\circ \)) and its implications for mathematical knowledge.

Examples

Example 1: Using Pythagoras’s Theorem

• Find the hypotenuse \( c \) of a triangle where \( a = 6 \, \text{units} \) and \( b = 8 \, \text{units} \):
  • \( c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100 \).
  • \( c = \sqrt{100} = 10 \, \text{units} \).

Example 2: Angles of Elevation and Depression

• A person standing on a cliff observes a boat at an angle of depression of \( 30^\circ \). If the cliff height is 50 meters, find the horizontal distance to the boat:
  • \( \tan 30^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{50}{x} \).
  • \( x = \frac{50}{\tan 30^\circ} = \frac{50}{0.577} \approx 86.6 \, \text{meters} \).

Example 3: Bearings and Navigation

• A ship travels 20 km on a bearing of \( 045^\circ \). How far north and east has the ship traveled?
  • North component: \( 20 \cos 45^\circ = 20 \cdot 0.707 \approx 14.14 \, \text{km} \).
  • East component: \( 20 \sin 45^\circ = 20 \cdot 0.707 \approx 14.14 \, \text{km} \).