For example, Trigonometry is used to analyse the angle. As like how modern algebra is used in reality and where it is applicable ?
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modern algebra is used in all scientific fields on almost a daily basis.  Its used everyday by the financial world in calculating interest and money values.  its used by the social sciences to study trends with society. A cashier at a store uses it when she rings up your order (the machine does it for them in most cases) Its basically used by everyone, everyday.
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Assume you have a number line, and through the origin (0) of that number line, you place another number line perpendicular to the first, so that the second origin is the same point as the first origin.  Now you have a [rectangular/Cartesian] coordinate plane.\nYou denote a point by measuring its distance along the horizontal axis from the origin, and then measuring its distance along the vertical axis from the origin and coming up with notation like (8,4).  It's 8 "to the right" and 4 "up" from the origin.\nWe use this plane to graph lines, circles, hyperbolas, parabolas, ellipses, etc.\n\nIf you don't want to get confused, stop reading here.\n\nThere is also a [circular/polar] coordinate plane, wherein you take a number line, and chop off half of it at the origin, so it's a ray.  The first coordinate is the distance from the origin along that ray.  The second coordinate is the measure of the angle you would make with the ray, and a second ray passing starting at the origin and including the point in question [typically measured in radians, so pi radians is 180°].  Like (4, 0) is 4 units "to the right" of the origin and (4,pi) is 4 units "to the left" of the origin.\nThis coordinate system is useful for graphing spirals and other looping curves.
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Angles associated with a Pyramid\n\ncheck out the site for the mathematical equation\n\n\n\n\n\n            In mathematics, the term pyramid refers to a rather general type of geometric solid.  One can find the definition here.  But the pyramids that one finds in different parts of the world generally have a rectangular base with the apex directly above the center of the base.  Most of the pyramids in Egypt actually have a square base and are therefore quite symmetrical. The four faces are isosceles triangles and are congruent to each other.  The shape of such a pyramid is determined by the angle that each face makes with the base, i.e. the angle formed by the horizontal line joining the midpoint of the base of a face to the center of the pyramid and the slanted line joining that midpoint to the apex of the pyramid.    That angle could be any angle between 0o and 90o. If that angle is close to 90o, then the pyramid would be very tall and would seem like a simple, pointed obelisk.  If that angle is close to 0o, then the pyramid would be almost flat and one would  probably be unhappy calling it a pyramid.  But the shape is also determined by various other quantities.  For example, the angle that each edge of the pyramid makes with the base would be sufficient.  (That angle is formed by the horizontal line joining a corner of the pyramid to the center and the edge joining that corner to the apex.)   Either of these angles is determined by its slope, or the inverse of its slope.  In modern Western Mathematics,  it would be customary to specify either angle by its slope, but the discussions of pyramids that one finds in the Egyptian Mathematical Papyri suggest that the Egyptian architect would have specified an angle by the inverse of its slope. For the slant-angle of the faces of a pyramid, the inverse-slope is referred to as the "seked" of the pyramid in the Rhind papyrus. \n\nhttp://www.math.washington.edu/~greenber/slope.html
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The basic idea behind trigonometry is that once you know the angles of a triangle, you'll know the ratios of the sides of the triangle.  And so if you know one side, you can use trig to find out the two other sides.\n\nSine, cosine, and tangent deal with right triangles.  They tell you the ratios of different pairs of sides in the right triangle.\n\nHere's a brief intro:\nhttp://opera.answers.com/trigonometry
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They don't have values, they are operations, which means each one tells you to do something different (and shame on your teacher for not explaining this to the class BEFORE having you do them on your calculator). \n\nYou always have to have right triangles when dealing with these three trig functions (that's what sine, cosine and tangent are, trig functions). They are each ratios, which means a comparison of two numbers. Sine means to take the length of the opposite side (the side opposite the angle you're measuring) and put that number in a fraction as the numerator, then in the denominator put the length of the hypotenuse of the triangle. So for short, sine means "opposite over hypotenuse." Similarly, cosine means "adjacent over hypotenuse" and tangent means "opposite over adjacent." The adjacent side is the side forming the angle that is NOT the hypotenuse (this is really difficult to explain without being able to draw you a picture). My students called it SOHCAHTOA for short to remember them all.\n\nI'll try and walk you through an example as best I can...Draw an isoceles right triangle (which means the triangle has one right angle and two 45 degree angles). Since it's isoceles, the two sides that are NOT the hypotenuse must be the same length. So let's say those two sides each have a length of 1. If you do the pythagorean theorem (a squared plus b squared equals c squared) you'll get that the length of the hypotenuse is equal to the square root of 2, or approximately 1.414. Did you draw the pictures and label all the parts? \n\nNow pick one of those 45 degree angles and circle it. Let's calculate the sine of 45 degrees, WITHOUT using a calculator. Using that angle you chose, find the length of the opposite side (the side of the triangle that isn't touching the angle at all). It's 1 right? So put that in a fraction as the numerator, the number on top. Sine means opposite over hypotenuse, so in the denominator we need the length of the hypotenuse, which is the square root of 2 or 1.414. So your equation now says....\n\nsin 45=1/1.414\n\nOn your calculator, punch in 1 divided by 1.414 and you'll get the number 0.70721....\n\nMake sure your calculator is in degree mode, if it's not you'll get the wrong answer, it's a very common mistake!\n\nNow punch in sin 45 and hit enter. What number do you get? 0.70710....\n\nThe reason the two numbers are different is because we rounded the length of the hypotenuse. If we'd actually used the square root function on our calculators and typed in the square root of 2 instead of the decimal, they would have been exactly the same number. \n\nAnd it doesn't matter what the lengths of the sides of the triangles are, as long as they're right triangles. IF YOU DON'T HAVE RIGHT TRIANGLES THOUGH, YOU CAN'T USE THESE FUNCTIONS. So be careful. But that's all the sine, cosine and tangent represent. They are the ratios of the side lengths of right triangles. \n\nHere's a final example. The tangent of 51 degrees (tan 51) equals 1.23489..... That means that in a right triangle with a 51 degree angle, if you take the length of the side opposite the 51 degree angle and divide it by the length of the side adjacent to that angle (remember, tangent means "opposite over adjacent"), then you'd get 1.23489....\n\nI hope that helped. Good luck! I'm sorry I couldn't draw you a picture.
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I believe the book is incorrect as well.\n\nThere are some real-world problems that start and end with numbers that do not belong to the set of Reals.  Consider the Quaternion.  A Quaternion is like a "super imaginary number", it is of the form, a + b*i + c*j + d*k, where i^2 = j^2 = k^2 = -1\n\nThis type of number can be used to represent the position and orientation of an object in 3D.  Since objects do have positions and orientations, it is not unreasonable to represent this with a Quaternion (even though many people don't know what it is).
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It depends on which leg you know the length of.  If it's the "adjacent"leg--the one that, along with the hypotenuse, forms the angle--then remember that the cosine of an angle is defined as the length of the adjacent angle divided by the length of the hypotenuse.\n\nIf it's the "opposite" leg--the one that isn't part of the angle--then you use the fact that the sine is defined as the length of the opposite leg divided by the length of the hypotenuse.\n\nEither way, you use the inverse trig functions on your calculator to find the angle.
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U should find out old class 9 Mathe. It can be helpful to u to understand easily basic of trigonometry i.e., 'Ө' and 'sine' in ur class standard.\nOld class 9 book can be available in old book shop. Find it at ur nearest old book shop.
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For the first part, use the fact that tan^2(x) = sin^2(x)/cos^2(x) and sec^2(x) = 1/cos^2(x) to transform your expression into (sin^2(x)-1)/(cos^2(x)). Then take the well-known identity sin^2(x) + cos^2(x) = 1 and re-write it to get sin^2(x)-1 = -cos^2(x). You should get -1 in the end.\n\nOr you could start with sin^2(x)+cos^2(x)=1, divide all terms by cos^2(x) and solve for tan^2(x)-sec^2(x). You should get -1 here as well.\n\nTo get sin(2a) use the double-angle formula:\nsin(2a)=2sin(a)cos(a)\nYou've got sin(a) but not cos(a), so draw the angle in the second quadrant whose sine is 5/13. This should give you cos(a) = -12/13 (you're going to have the ever famous 5-12-13 right triangle).\n\nFor cos(75), 75 degrees isn't an easy to use angle measurement but 150 degrees is. So use the half-angle measurements to write the cos(75) in terms of cos(150), which you can compute by drawing the 150 degree angle in the second quadrant and using ratios of 30-60-90 right triangles.\n\nHalf-angle formula:\ncos(a) = sqrt((1+cos(2a))/2)\nLet a = 75 to get cos(75) in terms of cos(150), figure out cos(150) from the appropriate 30-60-90 right triangle and plug it in.
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It depends on the school and how advanced the student is in their aptitude and experience. Advanced algebra, geometry and trigonometry are all fairly common. Occasionally you see some calculus for some students.
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In calculus and other higher math, angles come out naturally in radians, and the special values of radians (quarter circle, half circle, etc) are fractional values of pi (pi/2, pi, etc)\n\nThe graphs are just to get an idea in your mind how these functions look and are related to each other.\n\nAs a sixteen year old, you have lots of time to become acquainted with all this good stuff!
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the Cartesian coordinate system is develop used to uniquely determine each point in the plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.\nCartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. This work was influential in the development of analytic geometry, calculus, and cartography.\n\nThe idea of this system was developed in 1637 in two writings by Descartes. In part two of his Discourse on Method he introduces the new idea of specifying the position of a point or object on a surface, using two intersecting axes as measuring guides. In La Géométrie, he further explores the above-mentioned concepts.
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Can you give angle measures ?  Cant do much without those or some relationship between the other sides.
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Matrix algebra, a part of abstract algebra, is used to solve large system of equations. This is the basis for things various kinds of mathematical modeling programs such as economic forcasting, manufaturing production control, navigtation systems, etc.\n\nAnother areas where abstract algebra is used is in cryptography, i.e. generating encryption algorithms that protect your credit card number when you send it over the internet.\n\nWhat direct benefit you have from the knowledged will depend on what you plan to do with your future.
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The trigonometric identities have as their primary purpose to simplify calculations.  For example, why evaluate 2sinxcosx when you can evaluate sin2x.\n\nAs for the applications, oblique triangles are used in surveying and mapping.  For example, suppose you want to know your distance from an object on the opposite side of a river.  You can mark off a line of a fixed distance, say 100ft.  Then, go to the opposite ends of the line and use a transit to measure the angle between the object and the line you measured.  You can now determine all three sides with this information.
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Get a book from your library on applied math.\n\nInternate must be full of such applictions.
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algebra in vry simple word mean addition and subtraction n all sort of calculation can be done using these basic method.\n\nman  first learn to count which is just addition .\nso he learnt algebra first.  hence algebra came before geometry