Trigonometry is a branch of applied mathematics concerned with the relationship between angles and their sides and the calculations based on them. First developed as a branch of geometry focusing on triangles during the third century B.C., trigonometry was used extensively for astronomical measurements. The major trigonometric functions, including sine, cosine, and tangent, were first defined as ratios of sides in a right triangle. Since trigonometric functions are intrinsically related, they can be used to determine the dimensions of any triangle given limited information. In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit circle. This allowed the development of graphs of functions related to the angles they represent which were periodic. Today, using the periodic nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.
The word trigonometry stems from the Greek words trigonon, which means triangle, and metrein, which means to measure. It began as a branch of geometry and was utilized extensively by early Greek mathematicians to determine unknown distances. The most notable examples are the use by Aristarchus (310-250 B.C.) to determine the distance to the Moon and Sun, and by Eratosthenes (c. 276-195 B.C.) to calculate the Earth's circumference. The general principles of trigonometry were formulated by the Greek astronomer, Hipparchus of Nicaea (active 162-127 B.C.), who is generally credited as the founder of trigonometry. His ideas were worked out by Ptolemy of Alexandria (c. 90-168 A.D.), who used them to develop the influential Ptolemaic theory of astronomy. Much of the information we know about the work of Hipparchus and Ptolemy comes from Ptolemy's compendium The Almagest written around 150.
Trigonometry was initially considered a field of the science of astronomy. It was later established as a separate branch of mathematics, largely through the work of the mathematicians Johann Bernoulli (1667-1748) and Leonhard Euler (1707-1783).
Central to the study of trigonometry is the concept of an angle. An angle is defined as a geometric figure created by two lines drawn from the same point, known as the vertex. The lines are called the sides of an angle and their length is one defining characteristic of an angle. Another characteristic of an angle is its measurement or magnitude, which is determined by the amount of rotation, around the vertex, required to transpose one side on top of the other. If one side is rotated completely around the point, the distance travelled is known as a revolution and the path it traces is a circle.
Angle measurements are typically given in units of degrees or radians. The unit of degrees, invented by the ancient Babylonians, divides one revolution into 360° (degrees). Angles which are greater than 360° represent a magnitude greater than one revolution. Radian units, which relate angle size to the radius of the circle formed by one revolution, divide a revolution into 2 units. For most theoretical trigonometric work, the radian is the primary unit of angle measurement.
The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to 180°. The three points of a triangle, known as its vertices, are usually denoted by capital letters.
Triangles can be classified by the lengths of their sides or magnitude of their angles. Isosceles triangles have two equal sides and two congruent (equal) angles. Equilateral, or equiangular, triangles have three equal sides and angles. If no sides are equal, the triangle is a scalene triangle. All of the angles in an acute triangle are less than 90° and at least one of the angles in an obtuse triangle is greater than 90°. Triangles, such as these, which do not contain a 90° angle, are generally known as oblique triangles. Right triangles, the most important ones to trigonometry, are those which contain one 90°angle.
Triangles which have proportional sides and congruent angles are called similar triangles. The concept of similar triangles, one of the basic insights in trigonometry, allows us to determine the length of a side of one triangle if we know the length of certain sides of the other triangle. For example, if we wanted to know the height of a tree, we could use the idea of similar triangles to find it without actually having to measure it. Suppose a person is 6 ft (183 cm) tall and casts an 8 ft (2.44 m) long shadow. The tree, whose height is unknown, casts a shadow that is 20 ft (6.1 m) long. The triangles that could be drawn using the shadows and objects as sides are similar. Since the sides of similar triangles are proportional, the height of the tree is determined by setting up the mathematical equality: "height of tree" ÷ "length of tree's shadow" = "height of person" ÷ length of person's shadow" = x/20 = 6/8. By solving this equation, the height of the tree is found to be 15 ft (4.57 m).
The triangles used in the previous example were right triangles. During the development of trigonometry, the parts of a right triangle were given certain names. The longest side of the triangle, which is directly across from the right angle, is known as the hypotenuse. The sides which form the right angle, denoted by a box in the diagram, are the legs of the triangle. For either acute angle in the triangle, the leg that forms the angle with the hypotenuse is known as the adjacent side. The side across from this angle is known as the opposite side. Typically, the length of each side is denoted by a lower case letter. In the diagram of triangle ABC, the length of the hypotenuse is indicated by c, the adjacent side is represented by b, and the opposite side by a. The angle of interest is usually represented by .
The ratios of the sides of a right triangle to each other are dependent on the magnitude of its acute angles. In mathematics, whenever one value depends on some other value, the relationship is known as a function. Therefore, the ratios in a right triangle are trigonometric functions of its acute angles. Since these relationships are of most importance in trigonometry, they are given special names. The ratio or number obtained by dividing the length of the opposite side by the hypotenuse is known as the sine of the angle (abbreviated sin ). The ratio of the adjacent side to the hypotenuse is called the cosine of the angle (abbreviated cos ). Finally, the ratio of the opposite side to the adjacent side is called the tangent of , or tan . In the triangle ABC, the trigonometric functions are represented by the following equations: sin = a/c, cos = b/c, tan = a/b, sin /cos .
These ratios represent the fundamental functions of trigonometry and should be committed to memory. Many mnemonic devices have been developed to help people remember the names of the functions and the ratios they represent. One of the easiest is the phrase "SOH, CAH, TOA." This means: sine is the opposite over the hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
In addition to the three fundamental functions, three reciprocal functions are also defined. The inverse of sin , or 1/sin , is known as the secant of the angle or sec . The inverse of the cos is the cosecant or csc . Finally, the inverse of the tangent is called the cotangent of cot . These functions are typically used in special instances.
The values of the trigonometric functions can be found in various ways. They can often be looked up in tables which have been compiled over the years. They can also be determined by using infinite series formulas. Conveniently, most calculators and computers have the values of trigonometric functions preprogrammed in.
One immediate application for trigonometric functions is the simple determination of the dimensions of a right triangle, also known as the solution of a triangle, when only a few are known. For example, if the sides of a right triangle are known, then the magnitude of both acute angles can be found. Suppose we have a right triangle whose sides are 2 in (5 cm) and 4.7 in (12 cm), and whose hypotenuse is 5.1 in (13 cm). The unknown angles could be found by using any trigonometric function. Since the sine of one of the angles is equal to the length of the opposite side divided by the hypotenuse, this angle can be determined. The sine of one angle is 5/13, or 0.385. With the help of a trigonometric function table or calculator, it will be found that the angle which has a sine of 0.385 is 22.6°. Using the fact that the sum of the angles in a triangle is 180°, we can establish that the other angle is 180° - 90° - 22.6° = 67.4°.
In addition to solving a right triangle, trigonometric functions can also be used in the determination of the area when given only limited information. The standard method of finding the area of a triangle is by using the formula, area = 1/2b (base) x h(altitude). Often, the altitude of a triangle is not known, but the sides and an angle are known. Using the side-angle-side (SAS) theorem, the formula for the area of a triangle then becomes, area = 1/2 (one side) x (another side) x (sine of the included angle). For a triangle with sides of 5 cm and 3 cm respectively and an included angle of 60°, the area of the triangle would be equal to 1/2 x 5 x 3 x sin 60° = 13 cm2.
The formula for the area of a triangle leads to an important concept in trigonometry known as the Law of Sines which says that for any triangle, the sine of each angle is proportional to the opposite its opposite side, symbolically written: in triangle ABC, sin A ÷ a = sin B ÷ b = sin C ÷ c.
Using the Law of Sines, we can solve any triangle if we know the length of one side and magnitude of two angles, or two sides and one angle. Suppose we have a triangle with angles of 45° and 70°, and an included side of 15.7 in (40 cm). The third angle is found to be 180° - 45° - 70° = 65°. The unknown sides, x and y, are found with the Law of Sines because sin 45 ÷ x = sin 70 ÷ y = sin 65 ÷ 40.
The lengths of the unknown sides are then x = 12.29 in (31.2 cm) and y = 16.35 in (41.5 cm).
The Law of Sines can not be used to solve a triangle unless at least one angle is known. However, a triangle can be solved if only the sides are known by using the Law of Cosines which is stated in triangle ABC, c2 = a2 + b2 - 2ab cos C, or can be written
cos C = (a2 + b2 - c2) ÷ 2ab
which is more convenient when using only the sides to solve a triangle. As an example, consider a triangle with sides equal to 2 in, 3.5 in, and 3.9 in (5 cm, 9 cm, and 10 cm). The cosine of one angle would be equal to (52 + 92 - 102)/(2 x 59) = 0.067, which corresponds to the angle 86.2°. Similarly, the other two angles are found to be 29.9°and 63.9°.
In addition to the reciprocal relationships of certain trigonometric functions, two other types of relationships exist. These relationships, known as trigonometric identities, include cofunctional relationships and Pythagorean relationships. Cofunctional relationships relate functions by their complementary angles. Pythagorean relationships relate functions by application of the Pythagorean theorem.
The sine and cosine of an angle are considered cofunctions, as are the secant and cosecant, and the tangent and cotangent.
The Pythagorean Theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For a triangle with sides of x and y and a hypotenuse of z, the equation for the Pythagorean Theorem is x2+ y2 = z2. Applying this theorem to the trigonometric functions of an angle, we find that sin2 + cos2 = 1. Similarly, 1 + tan2 = sec2 and 1 + cot2 = csc2 . The terms such as sin2 or tan2 traditionally have meant (sin) x (sin) or (tan) x (tan).
In some instances, it is desirable to know the trigonometric function of the sum or difference of two angles. If we have two unknown angles, and , then sin ( + ) is equal to sincos + cossin. In a similar manner, their difference, sin(-) is sincos - cossin. Equations for determining the sum or differences of the cosine and tangent also exist and can be stated as follows.
cos( ± ) = coscos ± sinsin tan ( ± ) = (tan ± tan )/(1 ± tantan) These relationships can be used to develop formulas for double angles and half angles. Therefore, the sin 2 = 2sincos and cos 2 = 2cos2 - 1 which could also be written cos 2 = 1 - 2sin2.
For hundreds of years, trigonometry was only considered useful for determining sides and angles of a triangle. However, when mathematicians developed more general definitions for sine, cosine and tangent, trigonometry became much more important in mathematics and science alike. The general definitions for the trigonometric functions were developed by considering these values as points on a unit circle.
A unit circle is one which has a radius of one unit which means x2 + y2 = 1. If we consider the circle to represent the rotation of a side of an angle, then the trigonometric functions can be defined by the x and y coordinates of the point of rotation. For example, coordinates of point P(x,y) can be used to define a right triangle with a hypotenuse of length r. The trigonometric functions could then be represented by these equations: sin = y/r; cos = x/r; tan = y/x.
With the trigonometric functions defined as such, a graph of each can be developed by plotting its value versus the magnitude of the angle it represents.
Since the value for x and y can never be greater than one on a unit circle, the range for the sine and cosine graphs is between 1 and -1. The magnitude of an angle can be any real number, so the domain of the graphs is all real numbers. (Angles which are greater than 360° or 2 radians represent an angle with more than one revolution of rotation). The sine and cosine graphs are periodic because they repeat their values, or have a period, every 360° or 2 radians. They also have an amplitude of one which is defined as half the difference between the maximum (1) and minimum (-1) values.
Graphs of the other trigonometric functions are possible. Of these, the most important is the graph of the tangent function. Like the sine and cosine graphs, the tangent function is periodic, but it has a period of 180° or radians. Since the tangent is equal to y/x, its range is - to and its amplitude is .
The periodicity of trigonometric functions is more important to modern trigonometry than the ratios they represent. Mathematicians and scientists are now able to describe many types of natural phenomena which reoccur periodically with trigonometric functions. For example, the times of sunsets, sunrises, and comets can all be calculated thanks to trigonometric functions. Also, they can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound.
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