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# Trigonometry

150 150 Deborah

### Overview

Trigonometry in the form of triangulation is at the heart of navigation, whether it is by land, sea, or air. GPS systems use triangulation to find and fix positions, extrapolating what is unknown from what is known. Triangulation is also used to measure the distances between earth and distant stars and galaxies, thus aiding future space travel.

### Trigonometry on Land

The ancient Greeks, Egyptians, and other ancient civilizations developed methods to measure triangles accurately. The Egyptians used principles of trigonometry to build the Pyramids, while the Greeks developed extensive geometric and trigonometric proofs and applied them to many surveying and navigational problems. If the measurement of two angles are known, the third can always be calculated. The tangent of a right triangle can also be used to calculate valuable distances such as the height of a tall tree or of a mountain.

### Trigonometry and Surveying

Surveyors use a special type of telescope called a theodolite. It is mounted on a tripod, and measures distances and angles in three-dimensional space. Mounting a telescope on a tripod allows the telescope to be rotated, making very precise measurements possible. Known distances and angles can then be triangulated for accurate mapmaking, road building, and other engineering applications. The procedure for triangulation is similar to measuring the height of a tree. Suppose one knows the accurate measure of baseline AB. The distance to point C can be used by measuring the angles for AC and BC very accurately and using the trigonometric ratios.

Navigation by sea is complicated by large distances without landmarks in open ocean. The principles of trigonometry and triangulation apply. For most of the time humanity has moved through water for long distances, the only landmarks have been the positions of the sun by day and the stars by night. Those angles and distances can be measured accurately by using devices such as the marine sextant (for angles) and the chronometer (for the exact time that measurements are taken). Navigation by sea is based upon spherical trigonometry. The exact position of a ship can be determined by the angle the celestial body makes with the horizon, measured at a precise time. The angle and precise time measurements are compared with tables of known values.

GPS is short for Global Positioning System. It has grown from an original network of 24 satellites to a network of over 30 satellites from the United States. Similar systems are under development from Russia, China, Japan, the European Union, and India, and many satellites are fully operational. Satellites orbit over the same locations every day, and emit signals continuously giving the exact time and their location. Triangulation with particular satellites allows for precise location mapping.

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150 150 Deborah

### Overview

The demonstration of parallax is as close as an observer’s own two eyes. It is the measurement of how the same object appears from two different points of view. Astronomers use the very small angles observed by parallax to estimate distances and relative motion of objects ranging from the sun, moon, and planets to distant stars.

### Parallax

The principles of triangulation are used to measure distances for all celestial objects. Imagine a circle around the earth with 2 observatories roughly at opposite sides of the earth from each other, such as Calgary, Canada and Pretoria, South Africa.  That is the baseline of the triangle. The apparent position of the moon in the sky will be at a different angle as seen from one observatory than from another at a different location. The moon is the closest celestial object to the earth, so the angle that can be estimated from its movement is the largest, at nearly one degree. According to the principles of trigonometry, if the baseline of the triangle is known, as well as the top angle (measured in fractions of arcseconds), the length of the long side can be estimated accurately.

### Solar Parallax

The principle of measuring solar parallax, or parallax to any of the other planets or asteroids in the solar system, uses a baseline measurement of the earth at opposite locations of its orbit. For example, apparent motion can be measured in January and in June. Then, similar calculations can be done to estimate the distance to the other celestial object.

### Stellar Parallax

The measurement of the parallax of stars outside the solar system uses such small angles that ancient astronomers could not measure them precisely enough. The parsec is a unit of measure that is based upon parallax. It is equal to about 3.26 light years, an unimaginable distance to astronomers in the 1600s and 1700s. Usable measurements of parallax weren’t possible until the middle of the 19th century.

### Applications Using Space Telescopes

The most accurate measurements of parallax are being made far from the obscuring atmosphere of earth. The Hipparcos satellite was launched in 1989, specifically to measure parallax to distant stars, up to about 1600 light-years away. Its companion Gaia mission measured the distances to over a billion stars. Observations from the Hubble Space Telescope can detect distances to stars over 10,000 light years away.

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150 150 Deborah

### Overview

There are infinitely many regular polygons. However, there are a fixed number of regular polyhedra, called Platonic solids. Mathematicians use the definitions of the regular polygons and the characteristics of solid objects to arrive at their conclusions.

### Characteristics of Regular Polyhedra

Regular polygons, such as equilateral triangles, squares, equilateral pentagons, hexagons, heptagons, octagons, and so on, have all sides and angles equal. There are an infinite number of them in two-dimensional space. In three-dimensional space, solids with regular polygon faces are called regular polyhedra.

### Triangular Faces

In order for a solid to have regular triangular faces, the vertices of the triangles must meet at a point to make a corner. If three triangles meet at each vertex, there are 4 faces, called a tetrahedron. If four triangles meet at each vertex, there are 8 faces, called an octahedron. If five triangles meet at each vertex, there are 20 faces, called an icosahedron. No more triangles can form a corner at a vertex.

### Square and Pentagonal Faces

There is only one regular solid made of squares, the cube. Only 3 squares can meet at each point to form a cube. A cube has 6 faces, a top, bottom, and 4 sides. A pentagon has five sides. A regular pentagon has angles that have the same measurement, with sides that are the same length. If three pentagons meet at each vertex, there are 12 faces, called a dodecahedron.

### Other Polygons and Other Dimensions

Mathematicians since Plato have shown that there are only those five regular polyhedra in classical three-dimensional space. In higher dimensions than 3, analogues to the polyhedra are called polytopes. Their faces also meet at vertices, and they are symmetrical along various axes. In the 19th century, a mathematician found that there were 6 regular polytopes in four dimensions. One of those figures is a four-dimensional cube, or a cube of a cube, called a tesseract, familiar to readers everywhere. (The cube of a cube of a cube is a penteract.) In dimensions higher than 4, there are only 3 regular polytopes, that correspond to the tetrahedron, cube, and octahedron. The n-dimensional tetrahedron is called a simplex. The n-dimensional cube is also called a hypercube, familiar in science fiction. The n-dimensional octahedron is called a cross-polytope with multiple cells.

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150 150 Deborah

### Overview

The unit circle has as its center the origin point of the Cartesian coordinates x and y, and has a radius 1. The trigonometric functions are also called circular functions, because they describe relationships between angles on the unit circle. We use different ways to describe trigonometry in order to see how the relationships and equations apply to different situations.

### Special Values of the Trigonometric Functions

When the trigonometric functions are defined as circular, the special values of trigonometric functions can be seen on the unit circle. If t, the length of the arc from the coordinates (1, 0) equals 0, then sin t equals 0, cos t equals 1, and tangent t equals 0. This is because sin t equals y, cos t equals x, and tan t equals y/x. When t equals 0, y/x equals 0/1. If t equals π/4, then its coordinates are ([√2]/2, [√2]/2). Therefore, sin t is (√2)/2, cos t is (√2)/2, and tangent t is 1.

### Domains of the Trigonometric Functions

If a circular function results in division by zero, it is undefined. Therefore, each function has a specific domain. Sines and cosines are defined for all real values of t. Tangents and secants are defined for all real numbers, except when the value of x is 0. This happens when the value of t equals (π/2) + nπ. Similarly, cosecants (csc) and cotangents (cot), are defined for all real numbers other than nπ, which happens when x equals 0.

### Sign of the Trigonometric Functions

All other values of the trigonometric functions can also be seen on the unit circle. The unit circle is divided into quadrants by the Cartesian coordinates, so the signs of each circular function can be determined by the value of t. If the value of t has a positive value for both x and y, then it lies in Quadrant I. In Quadrant II, only the value of y is positive, and x has a negative value, so sin t and cosecant t will be positive. All other functions will be negative. In Quadrant III, both coordinates for x and y are negative. This means that tangents and their reciprocal cotangents will be positive. In Quadrant 4, x is positive and y is negative, so cosines and their reciprocal, secants, will be positive. There are several ways to remember the direction of the signs.

### Fundamental Identities

The trigonometric functions are related to one another because the formula for the unit circle is x2+ y2=1. Therefore, the cosecant of t (csc t) is defined as 1/sin t; the secant of t (sec t) as 1/cos t, and the cotangent of t (cot t) as 1/tan t. Similarly, sin2t + cos2t = 1, tan2t +1 = sec2t, and 1 + cot2t = csc2t.

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150 150 Deborah

### Overview

Circular trigonometric functions can be applied to situations in physical, biological, and social sciences involving data that follows a pattern that is not linear. Many of those patterns are periodic, and can be modeled by approximations of sine, cosine, or other functions.

### The Sine and Cosine Waves

The numerical value of t around the unit circle has the same value whether it is represented by 2π or τ, 4π or 2τ, and so on. Therefore, a sine function can be graphed as sin (t +2nπ) = sin t and a cosine function can be graphed as cos (t +2nπ) = cos t, for any real integer n. Both the sine wave and the cosine wave are periodic and repeat their graphs symmetrically.

### Amplitude, Period, and Phase Shift

The period is the time it takes for a complete cycle through all values, from the beginning through the highest point to the baseline down to the lowest point, then back to baseline again. The amplitude is the highest point of the t curve and the absolute value of the lowest point, often represented by the letter a. A curve can be shifted horizontally by moving it left or right, called the phase shift. In trigonometric language, a sine wave can be represented by the expression y = a sin kx, and a cosine wave can be represented by the expression y = a cos kx.

### Oscillations

Many things are variable in nature, and follow a sine wave, a cosine wave, or a combination of different types of waves. Electromagnetic energy, whether infrared, visible light, UV light, X-rays, microwaves, or sound waves follow periodic oscillations and can be modeled by periodic oscillations. The period of variable stars can also be measured and calculated. The motion of a pendulum or electrical currents also can be measured using periodic functions.

### Other Periodic Functions

Behavior that can be modeled by sine, cosine, and combinations of waves occurs in biology, physiology, and other social sciences. Populations of predators and prey tend to vary with a cyclical change that can be approximated by sine or cosine functions. The population of predators increase when more prey are available and decrease when fewer prey are available, so the populations mirror one another.

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150 150 Deborah

### Overview

Trigonometry is defined by the measurement of angles and their relationships. One of the ways that trigonometry can be applied is in the measurement of angles and circular motion.

### Angle Measure

Angles in trigonometry and calculus can be measured in radians, which is a relationship of the measurement of an angle by the arc it makes on a circle of radius 1. If the initial side of the ray is measured counterclockwise from the terminal side, it is a positive angle, and if the initial side of the ray is measured clockwise from the terminal side, it is a negative angle.

### Standard Position

The standard position for measuring an angle θ is to measure it with the vertex at the origin at the xy axis and the initial side of the angle along the x axis. This is the same concept as the angle at the unit circle. If two angles are both in the standard position, and both sides coincide, they are coterminal. The initial side of one angle is the terminal side of the other and the terminal side of one angle is the initial side of the other.

### Arcs and Sectors

The radius of the circle is one of the factors that determines the length s of the arc. Therefore, the radian measure θ of an angle equals the length of the arc s/radius r. Similarly, the area of a circular sector equals ½ r2θ. These values are only true when the angle θ is measured in radians, as those measurements are standardized for the radius of a circle.

### Circular Motion

Suppose a point is moving around the outside of a circle, for example, if an object is in a circular orbit in space. There are two ways to describe the rate that it is moving. The linear speed is the distance traveled divided by the elapsed time. The angular speed is the change in the measure of the central angle in radians divided by the elapsed time. In math language, the linear speed v equals the distance s/time t, or v=s/t. The angular speed is represented by the Greek letter omega ω, which equals the angle θ/time t, or ω =θ/t. Linear and angular speed are related such that linear speed v =radius r times angular speed ω, or v= rω. This is a very useful concept in circular mechanics and in orbital motion.

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150 150 Deborah

### Overview

Although many students are most familiar with the measurement of angles by degrees, there are other ways to measure angles. In calculus, advanced trigonometry, and applications of calculus to science, angles are measured in radians. The grad is a unit of angle measure used in surveying and as part of the metric system, and minutes and seconds of arc are used to measure angles for navigation and in astronomy.

### Degrees

Many people are familiar with measuring an angle by degrees, using a protractor. One degree is 1/360 of a circle, as a circle has 360 degrees. Angles that are less than 90o are acute, angles that are 90o are right angles, and angles that are more than 90o are obtuse. The measurement of a full circle as 360o stems from the Babylonians, who used multiples of 60 in their mathematical system.

Radians are less familiar. They measure the length of an arc divided by its radius. They are part of the SI system and used in many scientific applications, as well in mathematics. One radian is equal to 180/π degrees, so to convert degrees to radians, multiply the degree measure by π/180. To convert radians to degrees, multiply the radian measure by 180/π degrees. For example, 1 radian multiplied by 180/π equals about 57.296o. A 23o angle is about 0.401 rad.

### Arc Minutes

An arc minute is equal to 1/60th of one degree, and an arc second is equal to 1/60th of an arc minute. It is used in applications that involve very small angles, such as astronomy, optics, ophthalmology, optometry, and navigation.

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150 150 Deborah

### Overview

Trigonometric identities are relationships between trigonometric ratios that define them in terms of one another. They can be used to help solve problems that involve trigonometric functions.

### Reciprocal Identities

The reciprocal function to the sine is the cosecant, to the cosine is the secant, and to the tangent is the cotangent. In math language, the csc x = 1/sin x; sec x = 1/cos x, and cot x = 1/tan x. If tan x =sin x/cos x, then cot x also equals cos x/sin x. The reciprocal identities are helpful because all trigonometric ratios in a problem can be rewritten in terms of sines and cosines, which are often much easier to use.

### Pythagorean Identities

The Pythagorean identities are sin2x + cos2=1; tan2x +1 = sec2x; and 1 + cot2x = csc2x. Using the Pythagorean identities, one expression can be solved in terms of another by using substitution. The important thing to remember is that the ratios are squared ratios, similar to using the Pythagorean Theorem with right triangles.

### Using Known Information

One way to solve a trigonometric expression is to make sure that any rules of algebra are followed. For example, fractions can be put into common denominators and combined, using like denominators. Suppose that the fractions to be added are sin θ/cos θ +cos θ/ (1 + sin θ). In order to add the fractions, they must have common denominators. Therefore, (sin θ [1 +sin θ] +cos2θ)/ (cos θ (1 + sin θ) can be simplified to 1/cos θ, which is sec θ.

### Simplifying Trigonometric Expressions

Another key to simplifying trigonometric expressions is to write them in terms of sines and cosines, because they are often easier to use. Tables are written with values of sines and cosines. Also, many of the common formulas, such as the area of a triangle, express values using the sine of an angle, and the Law of Sines and the Law of Cosines can be used to solve values for the triangle. Applications are much easier to use when the trigonometric expression is simplified.

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150 150 Deborah

### Overview

Right triangles have special properties that are important to determine trigonometric ratios, such as sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot). Those ratios reflect the relationships between the opposite and adjacent angles of the right angle with the hypotenuse.

### Trigonometric Ratios

Suppose a right triangle has an angle θ for one of the acute angles. The sine of θ (sin θ) is the ratio of the opposite side to the hypotenuse, the cosine of θ (cos θ) is the ratio of the adjacent side to the hypotenuse, and the tangent of θ (tan θ) is the ratio of the opposite side to the adjacent side. Those three ratios can be measured by the mnemonic SOHCAHTOA. There are three other trigonometric ratios that are the inverses of sine, cosine, and tangent. The inverse of sine is the secant, and the secant of θ (sec θ) is the ratio of the hypotenuse to the opposite side. The inverse of cosine is the cosecant, and the cosecant of θ (csc θ) is the ratio of the hypotenuse to the adjacent side. The inverse of tangent is the cotangent, and the cotangent of θ (cot θ) is the ratio of the adjacent side to the opposite side. These relationships can be determined from the Pythagorean Theorem.

### Similar Right Triangles

All right triangles, regardless of size, are similar if they have angles that measure the same. Therefore, if one right triangle has an angle that measures θ, it will be similar to another right triangle that measures θ. The ratios sin, cos, tan, sec, csc, and cot will be the same for angle θ, no matter the length of the sides of the triangle.

### Special Right Triangles

Some right triangles have special measures and ratios that can also be deduced from the Pythagorean Theorem. They are the 45-45-90 triangle (also known as the π/4, π/4, and π/2 in radian measure), and the 30-60-90 triangle (also known as π/6, π/4, and π/2). They can be used to calculate the trigonometric ratios.

### Applications of Right Triangles

The principles behind right triangles and their ratios are used to calculate many different relationships, such as the height of a tree, a building, or a mountain, or the distance between the earth and the sun. A famous Greek geometer named Thales, who founded trigonometry, calculated the height of the Great Pyramid in Egypt, and Aristarchus estimated the distance from the earth to the sun.

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150 150 Deborah

### Overview

The Law of Cosines is true because of properties of any triangle, not just right triangles. In fact, the Pythagorean Theorem is a special case of the Law of Cosines. The Law of Cosines and the Law of Sines can be used together to solve triangles with trigonometric ratios, if enough facts are known about them.

### More Properties of Triangles

Any triangle has 6 pieces of information, covered by the three angles A, B, and C, and the lengths of the three opposite sides of the triangle a, b, c. By convention, side a is opposite angle A, side b is opposite angle B, and side C is opposite angle B. Also, the angles of any triangle add up to 180o, so if the measure of any two angles can be known or deduced from other properties of the figure, the measure of the third can be solved.

### The Law of Cosines

The Law of Cosines in words states that the squared measurement of any side of a triangle equals the sum of the squared measurement of the other two sides minus twice the product of their measurement multiplied by the cosine of that included angle. In math language, that means 3 separate equations: a2= b2+c2— 2bc cos A; b2= a2+c2—2ac cos B; and c2= a2+ b2—2ab cos C. Suppose that angle C is a right angle. Then, cos C is 0, and the equation becomes the Pythagorean Theorem a2 + b2= c2.

### Side-Angle-Side and Side-Side-Side

If the only three pieces of information that are known about a triangle are 2 of the sides and the angle between them, the Law of Cosines can be used to calculate one angle, and the Law of Sines can be used to calculate the other. In another application of the Law of Cosines, the angles of a triangle can be calculated if the measure of all three sides are known. That is because the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. The angles cannot be calculated using the measure of the sides alone by using the Law of Sines.

### The Area of a Triangle Revisited

The area of a triangle can be calculated by at least three different formulas, two of them using trigonometric functions. The most familiar formula is A =1/2 bh; or the area of a triangle A equals ½ the base times the height of the triangle. If there is the measure of an angle θ, the formula for area A becomes A =1/2 ab sin θ, where the measure of both sides a and b for included angle θ are all known. If the measure of all sides are known, the formula for the area of a triangle is the square root of the semiperimeter s (half the perimeter) times (s –a) (s-b) (s-c).

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