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Trigonometry 2
The height of a tower!
• Measure the horizontal distance to the tower
• Measure the angle from where you are standing to the top of the tower
• Apply trigonometry
• Distance 358 ft.
• Angle 23 degrees
Note: SOH-CAH-TOA
SOH-CAH-TOA
Calculations
• We need TOA!• We have Adjacent and the
Angle• TanA = 0.424 • Adjacent =358ft
• 0.424 = opposite/358ft.• Opposite = 358ft x 0.424• Opposite = 151.7ft• The height of the tower is
151ft.
Trig and Nav
• Distance between the Whalebone Arch and Whitby Abby is 575m
• Both are visible from the sea
Navigation using Landmarks
• Distance to land calculations
• Known landmarks
• Angle between landmarks
• Known distance between landmarks
• Whitby Whalebones and Whitby Abbey – distance 575m
• Measure the Angle and determine:
• How far away are you from land?
Note: SOH-CAH-TOA
First scenario: At right angle to the Abbey
• Distance Abbey to Whalebone Arch is 575m
• Ship at right angle to Abbey
• Angle to Whalebone arch is 25 degrees
• Note: SOH-CAH-TOA
Solution:
• We can use SOH to find the distance to the Arch
• Or TOA to find the distance to the Abbey and to Land
Distance to the Arch:Sin 25 degrees is 0.422Opposite = 575m0.422 = 575m/hypHyp = 575m/0.422Distance to Arch = 1362m
Distance to AbbeyTan 25 degrees is 0.466Opposite = 575m0.466 = 575m/ adjacentAdjacent = 575/0.466Adjacent = 1234mDistance to Abbey is 1234m
Using Universal Sin Rules
• Assume the ship is somewhere in the middle between the Abbey and the Whalebone Arch.
• We cannot use trigonometry which is designed for rectangular triangles
• There are only two options:1. Split a triangle in the middle and turn one triangle into two rectangular triangles2. Use Trigonometric Identities
Scenario
• Distance Abbey to Whalebone Arch is 575m
• Angle between Whalebone Arch and Abbey is 5 degrees
• Angle between nearest point of land and Abbey 2 degrees
• Angle between nearest point of land and Whalebone Arch is 3 degrees
Calculations
Distance from Abbey to nearest land point:
(575m/5) x 2 = 230m
Need to use TOA
Tan 2 = 230m/adjacent
0.0349 = 230 / adjacent
Adjacent = 230 / 0.0349
Distance to nearest land point = 6590m
Trigonometric Identities
• The law of Sines:
• The law of Cosines
Trigonometric Identities
• You will need to know two distances and one angle
• Or
• Two angles and one distance
Telephone Triangulation
• A phone has to be located
• Two towers pick up a signal and determine the direction
• The towers are located at a distance of 580m
• The direction of the signal is picked up from each tower 42 and 35 degrees
• Where is the phone located?
Calculations• Angle for the phone:
180 – 35 – 42 = 103 degrees
• a/sinA = b/sin B
Calcs for the first cell tower:
• 580m/sin 103 = b / sin 35
• (580 x sin35)/sin103 = b
• 332.6/0.974 = 341m
• The phone is at a distance of 341m from the first cell tower
Calcs. For the second cell tower:
(580 x sin42) / sin103 = c
388/0.974 = 398
The phone is at a distance of 398m from the 2nd cell tower