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This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools.
The introduction of the 4-6-2 design in 1901 has been described as "a veritable milestone in locomotive progress". [3] On many railways worldwide, Pacific steam locomotives provided the motive power for express passenger trains throughout much of the early to mid-20th century, before either being superseded by larger types in the late 1940s and 1950s, or replaced by electric or diesel-electric ...
AT&SF locomotive no. 1300, a 4-4-6-2 Mallet type. Under the Whyte notation for the classification of steam locomotive wheel arrangements, a 4-4-6-2 is a locomotive with two pairs of leading wheels, one set of four driving wheels, a second set of six driving wheels, and a pair of trailing wheels. Other equivalent classifications are:
Both of the Northern Pacific's 2-6-6-2 classes were copies of GN designs, including their Belpaire fireboxes, a rarity on NP steam locomotives. NP 3015, last of the Class Z's, had a troubled existence, suffering a crown sheet failure at Kennedy, Washington , on Stampede Pass circa 1916, then later derailing on the Wallace Branch in Idaho in 1933.
[4] [5] Although there were Ts and Ts-1 class 4-8-2 locomotives valid enough to that handle this work, they could not be relocated to some of SOU's routes over and over. [4] The SOU decided to revise the 4-6-2 type and ordered the more powerful Ps-4 Heavy Pacific class with the first batches built in 1923 by American Locomotive Company's (ALCO ...
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Chesapeake & Ohio Railway No. 1308 is an preserved articulated 2-6-6-2 "Mallet" type steam locomotive built by Baldwin Locomotive Works in 1949. It was the next to the last Class 1 mainline locomotive built by Baldwin, closing out more than 100 years of production, a total of more than 70,000 locomotives.