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8.1 Applications of trig functions (2d & 3d), other geometries  (Page 2/2)

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x 1 - x 2 + y 1 - y 2
Manhattan Distance (dotted and solid) compared to Euclidean Distance (dashed). In each case the Manhattan distance is 12 units, while the Euclidean distance is 36

The Manhattan distance changes if the coordinate system is rotated, but does not depend on the translation of the coordinate system or its reflection with respect to a coordinate axis.

Manhattan distance is also known as city block distance or taxi-cab distance. It is given these names because it is the shortest distance a car would drive in a city laid out in square blocks.

Taxicab geometry satisfies all of Euclid's axioms except for the side-angle-side axiom, as one can generate two triangles with two sides and the angle between them the same and have them not be congruent. In particular, the parallel postulate holds.

A circle in taxicab geometry consists of those points that are a fixed Manhattan distance from the center. These circles are squares whose sides make a 45 ∘ angle with the coordinate axes.

The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form. The distance between two points in Euclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In non-Euclidean geometry, straight lines are replaced with geodesics. Geodesics on the sphere are the great circles (circles on the sphere whose centers are coincident with the center of the sphere).The shape of the Earth more closely resembles a flattened spheroid with extreme values for the radius of curvature, or arcradius, of 6335.437 km at the equator (vertically) and 6399.592 km at the poles, and having an average great-circle radius of 6372.795 km.

Summary of the trigonomertic rules and identities

Pythagorean Identity Cofunction Identities Ratio Identities
cos 2 θ + sin 2 θ = 1 sin ( 90 ∘ - θ ) = cos θ tan θ = sin θ cos θ
cos ( 90 ∘ - θ ) = sin θ
Odd/Even Identities Periodicity Identities Double angle Identities
sin ( - θ ) = - sin θ sin ( θ ± 360 ∘ ) = sin θ sin ( 2 θ ) = 2 sin θ cos θ
cos ( - θ ) = cos θ cos ( θ ± 360 ∘ ) = cos θ cos ( 2 θ ) = cos 2 θ - sin 2 θ
tan ( - θ ) = - tan θ tan ( θ ± 180 ∘ ) = tan θ cos ( 2 θ ) = 2 cos 2 θ - 1
tan ( 2 θ ) = 2 tan θ 1 - tan 2 θ
Addition/Subtraction Identities Area Rule Cosine rule
sin ( θ + φ ) = sin θ cos φ + cos θ sin φ Area = 1 2 b c sin A a 2 = b 2 + c 2 - 2 b c cos A
sin ( θ - φ ) = sin θ cos φ - cos θ sin φ Area = 1 2 a b sin C b 2 = a 2 + c 2 - 2 a c cos B
cos ( θ + φ ) = cos θ cos φ - sin θ sin φ A r e a = 1 2 a c sin B c 2 = a 2 + b 2 - 2 a b cos C
cos ( θ - φ ) = cos θ cos φ + sin θ sin φ
tan ( θ + φ ) = tan φ + tan θ 1 - tan θ tan φ
tan ( θ - φ ) = tan φ - tan θ 1 + tan θ tan φ
Sine Rule
sin A a = sin B b = sin C c

End of chapter exercises

Do the following without using a calculator.

  1. Suppose cos θ = 0 . 7 . Find cos 2 θ and cos 4 θ .
  2. If sin θ = 4 7 , again find cos 2 θ and cos 4 θ .
  3. Work out the following:
    1. cos 15 ∘
    2. cos 75 ∘
    3. tan 105 ∘
    4. cos 15 ∘
    5. cos 3 ∘ cos 42 ∘ - sin 3 ∘ sin 42 ∘
    6. 1 - 2 sin 2 ( 22 . 5 ∘ )
  4. Solve the following equations:
    1. cos 3 θ · cos θ - sin 3 θ · sin θ = - 1 2
    2. 3 sin θ = 2 cos 2 θ
  5. Prove the following identities
    1. sin 3 θ = 3 sin θ - sin 3 θ 4
    2. cos 2 α ( 1 - tan 2 α ) = cos 2 α
    3. 4 sin θ · cos θ · cos 2 θ = sin 4 θ
    4. 4 cos 3 x - 3 cos x = cos 3 x
    5. tan y = sin 2 y cos 2 y + 1
  6. (Challenge question!) If a + b + c = 180 ∘ , prove that
    sin 3 a + sin 3 b + sin 3 c = 3 cos ( a / 2 ) cos ( b / 2 ) cos ( c / 2 ) + cos ( 3 a / 2 ) cos ( 3 b / 2 ) cos ( 3 c / 2 )
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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