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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 ...
In mathematics, the values of the trigonometric functions can be expressed approximately, as in (/), or exactly, as in (/) = /.While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots.
The symbol means that the ratio of the left-hand side and the right-hand side tends to one as . The symbol ≃ {\displaystyle \simeq } means that the difference between the left-hand side and the right-hand side tends to zero as n → ∞ {\displaystyle n\to \infty } .
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Euler's identity asserts that is equal to −1. The expression e i π {\displaystyle e^{i\pi }} is a special case of the expression e z {\displaystyle e^{z}} , where z is any complex number . In general, e z {\displaystyle e^{z}} is defined for complex z by extending one of the definitions of the exponential function from real exponents to ...
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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 ]
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