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