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A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians. [2] Since sin ( x ) = cos ( x − π / 2 ) , {\displaystyle \sin(x)=\cos(x-\pi /2),} the case of a sine can be omitted from this definition.
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.
The formula is given in verses 17–19, chapter VII, Mahabhaskariya of Bhāskara I. A translation of the verses is given below: [3] (Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees).
[1] [2] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision. There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions.
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 ...
In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...
Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6. Next, he proved a corresponding result for the sine using the trigonometric identity sin(θ) = cos(θ − π/2). [4] In 1956, Niven extended Lehmer's result to the other trigonometric functions. [2]
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.