Math127 Right Triangle Trig Applications

Applications of Right Triangle Trigonometry

Determining Heights and Lengths

• Example 1: Ladder Against Building

  • Given:

    • Ladder forms an angle of 72 degrees with the ground

    • Distance from ladder to building = 6 feet

  • Finding Height (a)

    • Use the tangent function:

      • Formula: ( \tan(72^\circ) = \frac{a}{6} )

      • Multiply both sides by 6:

        • ( a = 6 \times \tan(72^\circ) )

    • Calculation:

      • ( a \approx 18.47 ) feet

      • Height of building: 18.47 feet

  • Finding Length of Ladder (b)

    • Use the cosine function:

      • Formula: ( \cos(72^\circ) = \frac{6}{b} )

      • Rearrangement:

        • ( b \times \cos(72^\circ) = 6 )

      • Solving for b:

        • ( b = \frac{6}{\cos(72^\circ)} )

    • Calculation:

      • ( b \approx 19.42 ) feet

      • Length of ladder: 19.42 feet

Finding the Height of a Kite

• Example 2: Kite Height

  • Given:

    • Kite held 4 feet above the ground

    • Angle from horizontal to kite = 42 degrees

    • Length of string = 120 feet

  • Finding Height Above Ground

    • Define height from ground as ( height = a + 4 ) feet

    • Use sine function for height (a):

      • Formula: ( \sin(42^\circ) = \frac{a}{120} )

      • Rearrangement:

        • ( a = 120 \times \sin(42^\circ) )

    • Calculation:

      • ( a \approx 80 ) feet

  • Total Height of Kite:

    • ( x = a + 4 = 80 + 4 = 84 ) feet

Understanding Bearings

• Bearings Basics

  • Bearings use a fixed vertical line (north/south) as the reference point.

  • To express a direction, specify the angle from north or south towards east or west.

• Example Bearings

  • Point O to Point A:

    • North 42 degrees East

  • Point O to Point B:

    • North 74 degrees West

  • Point O to Point C:

    • South 40 degrees West

  • Point O to Point D:

    • South 69 degrees East

Plane Travel Example

• Example 3: Plane Flight

  • Given:

    • Leaves airport at south 32 degrees west

    • Speed = 320 knots

    • Time = 1.2 hours

  • Total Nautical Miles Covered:

    • Distance = 320 knots ( \times 1.2 \text{ hours} = 384 \text{ nautical miles} )

  • Finding South and West Distances:

    • Use a right triangle:

      • Hypotenuse = 384 nautical miles

      • Angle = 32 degrees

  • Calculating South (a) and West (b) Distances:

    • South: ( a = 384 \times \cos(32^\circ) \approx 326 \text{ nautical miles} )

    • West: ( b = 384 \times \sin(32^\circ) \approx 203 \text{ nautical miles} )

Boat Travel Example

• Example 4: Boat Journey

  • Given:

    • Travels north 52 degrees at 6 mph for 1 hour

    • Makes 90-degree turn to north 38 degrees west for 2 hours

  • Distance during North Travel:

    • Distance = 6 miles

  • Distance during New Course:

    • Distance = 12 miles

  • Finding Total Distance from Marina (uses Pythagorean theorem):

    • ( a = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180} \approx 13.4 ) miles

  • Returning Bearing Calculation:

    • Find angle using tangent:

      • ( \tan \theta = \frac{12}{6} \Rightarrow \theta \approx 63.4^\circ )

    • Final bearing:

      • ( 63.4 - 52 = 11.4^\circ )

      • Bearing: South 11.4 degrees East