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Exploring Triangles on Maps: Real-World Implications & Calculations

Discovering nonexistent triangles on maps with missing angles and sides using trigonometry. Detailed examples of UAE and Gulf countries. Includes calculations for angles and area.

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Exploring Triangles on Maps: Real-World Implications & Calculations

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  1. Introduction: • We used the internet explorer to find triangles which do not exist in the real world but they do on the map at least on ours. • The map hasn’t giving us full information. • Once the map missed an angle and once it has missed a side. • Anyhow we had to use the law of cosines and the law of sine to find the missing angle / side.112

  2. On the 1st triangle • On the 1st triangle the picture shows a triangle formed by 3 emirates which are located in the Untied Arab Emirates. • The picture was taken from Google earth and the angles and sides are pin-point accurate. The 3 emirates had a great past especially Dubai will it started as a main port in the gulf and look at it now… • The other 2 emirates are RAK and Fujairah, RAK stands for Ras Al Khaimah, Fujairah had a busy past a very recent one with the protégés when they invaded them and finally it has been set free since that time.

  3. 1- Find angle (R) by using the law of cosines: because we have three sides and we want to find one angle. r2= d2+f2– 2(d)(f) Cos(R) 1312= 922+ 862 – 2(92)(86) Cos(R) 1312 – 922 – 862= –15824Cos(R) 1301= –15824Cos(R) 1301/(-15824)= -15824Cos(R) Cos(R)= -0.08221= Cos-1(-0.08221) (R)≈ 95° 3- Find angle (D):- D= 180 – (95+41) = 44° D = 44° 2- Find angle (F) by using the law of Sins, to find the other angles. Sin(F)/f = Sin(R)/r Sin(F°)/86 = Sin(95°)/131 Sin(F)= 86 Sin(95°) / 131= 0.65399 Sin(F)= Sin-1(0.65399) ≈ 41° (F) ≈ 41° 4- The area of the triangle:- Area=½(86)(92)Sin(95) Area≈ 3941 km2

  5. On the 2nd triangle • As we mentioned before the details are pin-point accurate and the information were taken from Google Earth. • Anyhow on this triangle the 3 points That were chosen are capital cities of 3 different countries which are SaudiaArabia, Kuwait, and Bahrain. • The 1st one Riyadah used to be a deserted small city in the middle of the desert but due to amazing leadings and developments the city has became something truly astonishing. • The 2nd is Kuwait City which is obviously the capital of Kuwait it has the recognizable tower which required modern architecture although it was made a long time ago. And not to mention Manamah it was a pretty much nothing until oil wealth helped the super fast growth of Bahrain.

  6. 1- Find angle (K) by using the law of cosine, because we have three sides. Sin(K)/k = Sin(M)/m Sin(K)/486 = Sin(66°)/707 486 Sin(66°) / 707 = 0.62798 K= Sin-1(0.62798) ≈ 39° Medina 2- Find angle (R) by using the law of sines, because we have two sides and an angle opposite on of them. Sin(39)/486 = Sin(R)/474.5 474.5 Sin(39)/486 = 0.6144 Sin-1(0.6144)≈ 38° 4- Find the area of the triangle:- Area= ½ (k)(m)sin(R) Area = ½ (486)(707)sin(38°) Area = 171801 sin(38°) Area= 105771.25km2 3- Find angle (M) by subtract (R) and (K) from 180°. M= 180 – (39+38) = 103°

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