"Spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere. Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation. The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and"
"both the length of the hypotenuse and the angle the hypotenuse makes with the base line (""c"" above) at the same time when given ""x"" and ""y"". The angle returned is normally given by atan2(""y"",""x""). Orthographic projections: By means of trigonometric ratios, one can obtain the value of two acute angles, formula_3 and formula_4, of the right triangle. Given the length of the hypotenuse formula_5 and of a cathetus formula_6, the ratio is: The trigonometric inverse function is: in which formula_9 is the angle opposite the cathetus formula_6. The adjacent angle of the catheti formula_6 is formula_3 = 90° –"
"Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse"
"formula_9 One may also obtain the value of the angle formula_9 by the equation: in which formula_16 is the other cathetus. Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the"
"the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes. The red section on the right, , is the difference between the lengths of the hypotenuse, , and the adjacent side, . As is shown, and are almost the same length, meaning is close to 1 and helps trim the red away. The opposite leg, , is approximately equal to the length of the blue arc, . Gathering facts from geometry, , from trigonometry, and , and from the picture, and leads to: Simplifying leaves, can also be expressed this way,"
"is the square root of 25, that is, 5. The word ""hypotenuse"" is derived from Greek (sc. or ), meaning ""[side] subtending the right angle"" (Apollodorus), The Greek term was loaned into Late Latin, as ""hypotēnūsa"". Adoption as a learned Latinism used in modern languages dates to the 16th century. The spelling in ""-e"", as ""hypotenuse"", is French in origin (Estienne de La Roche 1520). The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to"
"cosine, it follows that Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: formula_20 where formula_21 is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed. For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle ""θ"" with the horizontal, where ""θ"" may be computed as follows: The two-argument atan2 function computes the arctangent of ""y"" / ""x"" given ""y"" and ""x"", but with a range of (−,"
"angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length and . From this the trigonometric function tangent of 60° equals , and the sine of 60° and the cosine of 30° both equal . The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°. It is the distance between parallel sides of a regular hexagon with sides of length"
"a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle α, the opposite side, adjacent side and hypotenuse are labeled ""O"", ""A"" and ""H"" respectively, then the trigonometric functions are For the expression of hyperbolic functions as ratio of the sides of a"
"altitude to the hypotenuse is related to the legs of the right triangle by For solutions of this equation in integer values of ""a, b, f"", and ""c"", see here. The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter—the intersection of its three altitudes—coincides with the right-angled vertex. The Pythagorean theorem states that: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the"
"the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length ""c"", and the legs have lengths ""a"" and ""b"", then the theorem states that The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side ""c"". Some other facts about right triangles: For all triangles, angles and sides are related by the law of cosines and law of sines (also called the ""cosine rule"" and ""sine rule""). The triangle inequality states that the sum"
"and ""G"" and hypotenuse ""A"", then and where formula_40 is the golden ratio formula_41 Since the sides of this right triangle are in geometric progression, this is the Kepler triangle. Thales' theorem states that if ""A"" is any point of the circle with diameter ""BC"" (except ""B"" or ""C"" themselves) ""ABC"" is a right triangle where ""A"" is the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right"
"as shown in the upper part of the diagram, with ""BC"" the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ""y"", the side ""AC"" of length ""x"" and the side ""AB"" of length ""a"", as seen in the lower diagram part. If ""x"" is increased by a small amount ""dx"" by extending the side ""AC"" slightly to ""D"", then ""y"" also increases by ""dy"". These form two sides of a triangle, ""CDE"", which (with ""E"" chosen so ""CE"" is perpendicular to the hypotenuse) is a right triangle approximately similar to ""ABC""."
"and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse. In any right triangle the diameter of the incircle is less than half the hypotenuse, and more strongly it is less than or equal to the hypotenuse times formula_44 In a right triangle with legs ""a"", ""b"" and hypotenuse ""c"", with equality only in the isosceles case. If the altitude from the hypotenuse is denoted ""h"","
"Pythagorean theorem In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides ""a"", ""b"" and ""c"", often called the ""Pythagorean equation"": where ""c"" represents the length of the hypotenuse and ""a"" and ""b"" the lengths of the triangle's other two sides. Although it is"
"Hypot Hypot is a mathematical function defined to calculate the length of the hypotenuse of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers. Calculating the length of the hypotenuse of a triangle is possible using the square root function on the sum of two squares, but hypot(""x"", ""y"") avoids problems that occur when squaring very large or very small numbers. The magnitude of the hypotenuse from (0, 0) to (""x"", ""y"") can be calculated using: This operation is also known as Pythagorean addition. However the squares of very large or small"
"and ""b"" are the legs of the triangle. If the incircle is tangent to the hypotenuse AB at point P, then denoting the semi-perimeter as ""s"", we have and , and the area is given by This formula only applies to right triangles. If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangles which are both similar to the original and therefore similar to each other. From this: In equations, where ""a"", ""b"", ""c"", ""d"", ""e"", ""f"" are as shown in the diagram. Thus Moreover, the"
"proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I,"
"below, but this is known as the Pythagorean one. If ""c"" denotes the length of the hypotenuse and ""a"" and ""b"" denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: If the length of both ""a"" and ""b"" are known, then ""c"" can be calculated as If the length of the hypotenuse ""c"" and of one side (""a"" or ""b"") are known, then the length of the other side can be calculated as or The Pythagorean equation relates the sides of a right triangle in a simple way, so that if"
"hypotenuse to the length of the adjacent side, so called because it represents the secant line that ""cuts"" the circle (from Latin: ""secare"", to cut): The cosecant (""secant complement"", Latin: ""cosecans"", ""secans complementi"") of an angle is the reciprocal of its sine, that is, the ratio of the length of the hypotenuse to the length of the opposite side, so called because it is the secant of the complementary or co-angle: The cotangent (""tangent complement"", Latin: ""cotangens"", ""tangens complementi"") of an angle is the reciprocal of its tangent, that is, the ratio of the length of the adjacent side to"
"are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the ""definitions"" of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations. The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for"
"largest side is and the hypotenuse is 1. This yields: There are some commonly used special values in trigonometric functions, as shown in the following table. The symbol here represents the point at infinity on the projectively extended real line, the limit on the extended real line is on one side and on the other. The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function. Therefore, to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are"
"straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled ""1"") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is"
"summarized in the following table and described in more detail below. The angle is the angle between the hypotenuse and the adjacent line – the angle at ""A"" in the accompanying diagram. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin ""sinus"" for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle ""opens"". In our case: The cosine (""sine complement"", Latin: ""cosinus"", ""sinus complementi"") of an angle is the ratio of the"
"two solutions (""k"" = ±1) for the two external tangent lines: Geometrically this corresponds to computing the angle formed by the tangent lines and the line of centers, and then using that to rotate the equation for the line of centers to yield an equation for the tangent line. The angle is computed by computing the trigonometric functions of a right triangle whose vertices are the (external) homothetic center, a center of a circle, and a tangent point; the hypotenuse lies on the tangent line, the radius is opposite the angle, and the adjacent side lies on the line of"
"the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation."
"The acronym ""SOH-CAH-TOA"" is a useful mnemonic for these ratios. The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides. Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse. Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypotenuse. Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side. In introductory"
"all points (""x"", ""y"") such that This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |""x"" − ""a""| and |""y"" − ""b""|. If the circle is centred at the origin (0, 0), then the equation simplifies to Parametric form <br> The equation can be written in parametric form using the trigonometric functions sine and cosine as where ""t"" is a parametric variable in the range 0 to"
"where both the top of the tree and the base of the tree are above eye level, and where both the top of the tree and base of the tree are located below eye level. In the first situation, if D1 is the distance to the top of the tree as measured with a laser rangefinder, and (a) is the angle to the top of the tree measured with a clinometer, then this forms the hypotenuse of a right triangle with the base of the triangle at eye level. The height of the tree above eye level is [h1 ="
"it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse. ...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which"
"space. A discretized triangle, n units tall and n units long, can be constructed on the tiling. The hypotenuse of the resulting triangle will be n tiles long. However, by the pythagorean theorem, a corresponding triangle in a continuous space—a triangle whose height and length are n -- will have a hypotenuse measuring n√2 units long. To show that the former result does not converge to the latter for arbitrary values of n, one can examine the percent difference between the two results: = 1-. Since n cancels out, the two results never converge, even in the limit of large"
"removes the minus signs): All that remains is to prove it for this can be done by squaring the symmetry identities to get The identities and are also called Pythagorean trigonometric identities. If one leg of a right triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg, and the secant of the angle is the length of the hypotenuse. and: In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can"
"same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin ""θ"" and adjacent side of size cos ""θ"" in units of the hypotenuse. The Pythagorean theorem relates the cross product and dot product in a similar way: This can be seen from the definitions of the cross product and dot product, as with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity. This can also be used to"