### Trigonometry - New World Encyclopedia

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the This complex exponential function is sometimes denoted cis x ("cosine plus i sine "). The formula is still valid . Relationship between sine, cosine and exponential function. All of the following relationships need to be memorized. These are the most commonly used sin2 q + cos2 q = 1, 1 + tan2 q = sec2 q, 1 + cot2 q = csc2 q. Date: 04/30/ at From: Vaughn Wassmer Subject: Sine and Secant When someone applied the terms sine, cosine, tangent, secant, cosecant, names to the functions based only on their relation to the sine and cosine, we could.

And it's obvious now you point it out. So you can also construe the "co-" versions as exchanging opposite for adjacent edge, at least in the first quadrant.

## Trigonometry

After that we can define the three "co-" functions by exchanging the words "opposite" and "adjacent". As Steve says, going "co-" corresponds to shifting to the other acute angle in the right triangle.

But depending on which angle we choose as our angle of interest, what side counts as "adjacent" and which as "opposite" changes.

**Basic Trigonometry**

For negative angles or those beyond a right angle, it is necessary to consider the unit circle rather than SOHCAHTOA as a definition for the trigonometric functions image source: This shows how the three trigonometric identities above hold for angles which are not acute also. For instance, here the reflection of the sine graph blue gives us the cosine graph red. Since there is a reason to call it secant, there is no reason to call it the cosecant.

You are supposing that trig functions ought to be named in such a way that "co's" are reciprocals of other "co's". That is, the "co-something" is the "something" of the complement. That's how tangent and cotangent are related, though they also happen to be reciprocals. There is no naming convention that indicates which functions are reciprocals. But I'll suggest below that it really is more consistent than you realize.

The name "secant" refers to its representing the length of the secant line OB in Dr. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta inused an interpolation formula to compute values of sines, up to the second order of the Newton -Stirling interpolation formula. Khayyam solved the cubic equation and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle.

An approximate numerical solution was then found by interpolation in trigonometric tables. Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara inalong with some sine and cosine formulae. Bhaskara also developed spherical trigonometry. The thirteenth century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline.

Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry. In the fourteenth century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg grandson of Timur produced tables of trigonometric functions as part of their studies of astronomy.

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in which may have coined the word "trigonometry" itself. Overview In this right triangle: If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to degrees.

### Euler's formula - Wikipedia

The two acute angles therefore add up to 90 degrees: They are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle.

These ratios are given by the following trigonometric functions of the known angle A, where a, b, and c refer to the lengths of the sides in the accompanying figure: The sine function sindefined as the ratio of the side opposite the angle to the hypotenuse.

The cosine function cosdefined as the ratio of the adjacent leg to the hypotenuse. The tangent function tandefined as the ratio of the opposite leg to the adjacent leg.

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The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A.