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The Lutheran divine Martin Chemnitz, who contributed to the development of Lutheran systematic theology, delineated what are considered mortal sins: [49] For Scripture distinguishes between sins, namely that in the saints or reborn there are some sins because of which they are not condemned, but at the same time retain faith, the Holy Spirit ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
Chrysostom: For while as yet the sacrifice had not been offered, nor remission of sin sent, nor the Spirit had descended on the water, how could sin be forgiven? But since the Jews never perceived their own sin, and this was the cause of all their evils, John came to bring them to a sense of them by calling them to repentance. [10]
For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference". [45] He also discovered the law of sines for spherical trigonometry: [41]
The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ]
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [51] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [52]
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