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.

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.

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.

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.

The trigonometric functions are related to one another because the formula for the unit circle is x^{2}+ y^{2}=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, sin^{2}t + cos^{2}t = 1, tan^{2}t +1 = sec^{2}t, and 1 + cot^{2}t = csc^{2}t.

Interested in trigonometry tutoring services? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Independence, MO: visit: Tutoring in Independence, MO

]]>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 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.

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.

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.

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.

Interested in trigonometry tutoring services? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Billings, MT: visit: Tutoring in Billings, MT

]]>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.

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.

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.

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 r^{2}θ. These values are only true when the angle θ is measured in radians, as those measurements are standardized for the radius of a circle.

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.

Interested in trigonometry tutoring services? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in North Platte, NE: visit: Tutoring in North Platte, NE

]]>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.

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 90^{o }are acute, angles that are 90^{o }are right angles, and angles that are more than 90^{o }are obtuse. The measurement of a full circle as 360^{o} 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 57.296^{o}. A 23^{o }angle is about 0.401 rad.

Grads, short for gradian, are closely related to radians. A full turn or 2π is equal to 400 grads. It is an alternative measure of angles used in France and some other European countries, and it is used by surveyors. A 45^{o }angle is 50 grads, a 90^{o }angle is 100 grads, a 135^{o}angle is 150 grads, and a straight line is 180^{o }or 200 grads. In France, a common angle measure is the centigrad, or 1/100 of a grad. That term is so similar to the term for centigrade scale to measure temperature that the name of the temperature scale was changed to Celsius to honor its developer and avoid confusion.

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

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Lebanon, NH: visit: Tutoring in Lebanon, NH

]]>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.

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.

The Pythagorean identities are sin^{2}x + cos^{2}=1; tan^{2}x +1 = sec^{2}x; and 1 + cot^{2}x = csc^{2}x. 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.

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] +cos^{2}θ)/ (cos θ (1 + sin θ) can be simplified to 1/cos θ, which is sec θ.

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.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Buffalo, NY: visit: Tutoring in Buffalo, NY

]]>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.

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.

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.

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.

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.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Las Vegas, NV: visit: Tutoring in Las Vegas, NV

]]>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.

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 180^{o}, 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 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: a^{2}= b^{2}+c^{2}– 2bc cos A; b^{2}= a^{2}+c^{2}—2ac cos B; and c^{2}= a^{2}+ b^{2}—2ab cos C. Suppose that angle C is a right angle. Then, cos C is 0, and the equation becomes the Pythagorean Theorem a^{2 }+ b^{2}= c^{2}.

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 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).

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Portales, NM: visit: Tutoring in Portales, NM

]]>Some of the properties of a triangle hold true even if it is an oblique triangle, when none of the angles are right angles. We can solve the sides and angles of other triangles by using trigonometric ratios, as long as we have enough information.

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. In order to solve the triangle, at least 3 pieces of information must be known, and one of those must be the length of a side. Similarly, if we know the measure of 2 sides and the included angle, we can figure out the rest of the triangle. If we know the measure of all three angles, we cannot deduce the measure of the sides, however. The triangles may be similar, but not unique. In order to solve the triangle, there can only be one possibility.

In any triangle with angles A, B, C and opposite sides a, b, c, (sin A)/a = (sin B)/b = (sin C)/c. This is because the area of the triangle ABC can be expressed by the measurement of its sines times the length of two sides, no matter what sides and angles are chosen.

If we know the measure of 2 angles, we can find the measure of the third, because the degree measure of the angles of a triangle add up to be 180^{o}. That will make it possible to use the Law of Sines. This is always true whether the side is the initial side of both of the known angles or the terminal side of one of the angles. The known information forms only one triangle.

If two sides are known and one angle, there are three possibilities. Sometimes there is only one triangle with the information. There can be two possibilities that fit the data. Sometimes, the data presents an impossible situation. No triangle can be constructed to fit the situation.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Lyndhurst, NJ: visit: Tutoring in Lyndhurst, NJ

]]>The trigonometric functions can also be defined in terms of the unit circle, so sometimes they are called the circular functions. This is because of the relationships of the trigonometric functions of angles to the unit circle. The trigonometric functions correspond to coordinates on the unit circle, also called terminal points.

Locations along the unit circle are real numbers that can be expressed as x, y coordinates. Relationships between those numbers can be expressed as trigonometric functions. Suppose that t is a value on the unit circle, representing its distance from the (1, 0) coordinate. The terminal point (x, y) can be defined in math language so that the sine of t (sin t) equals y, the cosine of t (cos t) equals x, and the tangent of t (tan t) equals y/x, when x is not equal to 0. (Dividing by zero is undefined and is not allowed.)

The other trigonometric functions are also circular functions. The cosecant (csc) is the reciprocal of the sine, so that the cosecant of t equals 1/y, when y is not equal to 0. The secant (sec) is the reciprocal of the cosine, so that sec t is 1/x, when x is not equal to 0. The cotangent (cot) is the reciprocal of the tangent, so cot t equals x/y when y is not equal to 0.

The trigonometric functions of angles and trigonometric functions measured in terms of the unit circle measure the same thing. Suppose that a right triangle is placed in the coordinate plane so that the vertex of the angle to be measured (call it theta θ) is on the center of the circle. The adjacent side to the right angle is along the x axis. In this example, the hypotenuse is 1, equal to the radius of the unit circle, but the sides of the triangle itself could extend beyond the unit circle.

In terms of the right triangle, the trigonometric functions of the angle θ follow SOHCAHTOA. The sine of θ is opposite/hypotenuse, the cosine of θ is adjacent/hypotenuse, and the tangent of θ is opposite/adjacent. Note also that the arc between (0, 1) and (x, y) is equal to t, because it measures the same quantity.

Interested in pre-calculus tutoring services? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Natchez, MS: visit: Tutoring in Natchez, MS

]]>One of the methods of measuring angles is to measure them in relationship to a unit circle and as the measurement of an arc. Applications include periodic behavior such as light, sound or other waves along the electronic spectrum, and the processing of signals.

Suppose that a circle is constructed from a central point at the origin of the xy plane so that it has a radius of 1 unit. The equation for the unit circle is x^{2} + y^{2} = 1., which means that no matter what real numbers are chosen as values for x and y, they will fall at a particular point on the circumference of the circle, as long as the numbers satisfy the equation.

On the unit circle the origin is on values (0, 0); the circumference crosses the y axis at 2 points (1, 0; -1, 0); and the x axis at 2 points (0, 1; 0,-1). Any point P along the circumference can be measured by its distance t on the unit circle. The distance t is a real number. For values of t that are greater than 0, movement is counterclockwise from (1, 0), and for values of t that are less than 0, movement is clockwise from (1, 0). A full circumference of the circle is 2π (or tau τ).

Another way of looking at terminal points around the unit circle is that the distance t is a real number that is the same as an arc. Therefore, a full arc is once around the circle (2π or τ), a half-arc is halfway around the circle. Different values of t are represented by the same terminal point.

Still another way to look at a terminal point on the unit circle is by using the same general description of its location as any other point on the coordinate plane. A reference number is the shortest distance along the circumference of a circle between the terminal point t and the x axis. They are used as a shortcut to find the coordinates of the terminal point and the quadrant where it lies. Thus a terminal point with a positive value for x and a positive value for y will lie in Quadrant I; a negative value for x and a positive value for y, Quadrant II; a negative value for both x and y, Quadrant III; and a positive value for x and a negative value for y, Quadrant IV.

Interested in pre-calculus tutoring services? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Mankato, MN: visit: Tutoring in Mankato, MN

]]>