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Introduction |
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Trigonometry is that branch of mathematics which deals with the measurement of sides and angles of triangles. |
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Angles |
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An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position. The important terms are: Measurement of angle, Positive and Negative angles, Lines at right angles, Quadrants, Angle in standard position. |
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Measurement of Angles |
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To measure an angle we shall use two kinds of units, the degree unit and the radian unit. |
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A degree(Sexagesimal system) is defined to be an angle formed by half-line (or a ray) rotated about its end point (1/360) of a complete revolution. |
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Types of Angles:
Acute angle, Obtuse angle, Right angle, Reflex angle, A straight angle. |
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Radian measure [Circular system] |
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A radian is the measure of an angle at the centre of a circle subtended by an arc, whose arc length is equal to the radius of the circle. |
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Grade measure (Centesimal system or French system) |
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The principal unit of this system is grade(g) which is one hundreds part of right angle.
The attempt to introduce this grade system in France was not successful. |
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Trigonometric Functions |
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The circle whose radius is 1 unit whose centre is the origin of a rectangular co-ordinate system is called the unit circle. |
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1. cosq = x.
2. sinq = y.
3. tanq = y/x.
4. secq = 1/x.
5. cosecq = 1/y.
6. cotq = x/y.
The six functions of q defined by the above equation are called trigonometric functions of q or circular functions of q. |
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Circular Functions |
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The circular functions are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. |
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Periodic Functions |
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The trigonometric functions belong to a large class of functions called periodic functions in which there is a regular repetition of the values of the function over a certain interval. |
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The variation of the values |
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As the Sine of an angle increases the Cosine of the angle decreases from the value 0 to 1 and vice versa. |
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Even and Odd functions |
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i) A function f(x) is said to be even if f(-x) = f(x). |
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ii) A function f(x) is said to be odd if f(-x) = -f(x). |
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Tangent Functions |
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The tangent function is defined in the form as tanq = x/y, where y is not equal to zero. |
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Other Circular Functions |
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1. The reciprocal of cosq = 1/cosq = secq. |
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2. The reciprocal of sinq = 1/sinq = cosecq. |
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3. The reciprocal of tanq = 1/tanq = cotq. |
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Identical properties of circular functions and trigonometric functions |
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\ The Circular
function of a real number q and the
trigonometric function for the angle qc are same. |
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The identities for circular functions of real numbers hold for trigonometric functions of angle q. |
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Values of Trigonometric Functions |
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The Values of Trigonometric Functions of 90o and 0o are:
cos90o = 0, sin90o = 1, tan90o = Not defined, sec90o = Not defined, cot90o = 0, cosec90o = 1, cos0o = 1, sin0o = 0, here cosec0o and cot0o are not defined, sec0o = 1, tan0o = 0. |
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Signs of Trigonometric Ratios |
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i) The numerical values of sinq and cosq cannot be greater than 1. |
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ii) The numerical values of sec q and cosec q can never be less than 1. |
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iii) There is no restriction on the values of tanq and cotq since they can take any value. |
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Trigonometric ratios of Allied Angles |
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The following table shows the Trigonometric ratios of Allied Angles:
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- q |
90°- q |
90° + q |
180°- q |
180° + q |
270°- q |
270° + q |
2nP - q |
2nP + q |
| sin q |
-sin q |
cos q |
cos q |
sin q |
-sin q |
-cos q |
-cos q |
-sin q |
sin q |
| cos q |
cos q |
sin q |
-sin q |
-cos q |
-cos q |
-sin q |
sin q |
cos q |
cos q |
| tan q |
-tan q |
cot q |
-cot q |
-tan q |
tan q |
cot q |
-cot q |
-tan q |
tan q |
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Sum Formula |
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For any two angles A and B, we have |
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1. Sin(A+B) = SinA CosB + CosA SinB. |
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1. Cos(A+B) = CosA CosB + SinA SinB. |
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1. Tan(A+B) = (TanA + TanB) / (1 - TanA TanB). |
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1. Cot(A+B) = (CotB CotA - 1) / (CotB + CotA). |
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Trigonometric Ratios of Multiple and Sub-multiple Angles |
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Trigonometric Ratios of Multiple and Sub-multiple Angles are:
1. Sin2A = 2SinACosA
2. Cos2A = Cos2A-Sin2A. |
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Trigonometric Ratios of multiple and some special angles |
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Trigonometric Ratios of multiple and some special angles are:
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Conditional Trigonometric Identities |
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In the above topics many identities have been discussed. They are true for all values of the angles for which trigonometric functions are defined. In this section we prove identities, where a certain relationship exists among the angles considered. |
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Many interesting and important identities are established using the relation A+B+C=180. |
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Graphs |
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In plotting the graph of any trigonometric function the angle may be regarded as measured either in radians or in degrees. If a trigonometric function is combined with an algebraic function it is customary to assume that the angle is measure in radians. |
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Graph of y = sin x |
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As x increases from 180o to 270o, sinx decreases from 0 to -1 and as x increases from 270o to 360o, sinx increases from -1 to 0. The maximum absolute value of sin x = 1. |
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Graph of y = cosx |
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As x increases from 0o to 90o cosx decreases from 1 to 0, as x increases from 90o to 180o cosx decreases from 0 to -1, as x increases from 180o to 270o cosx increases from -1 to 0, as x increases from 270o to 360o cosx increases from 0 to 1. Cosx is period and has a period 2p. |
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Graph of y = tanx |
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the part of the curve from p to 2p has the same form on the part from 1 to p. This implies that the fact tan x = tan (p + x). The whole curve consists of an endless number of branches having the same form as the branch
corresponding to the values of x from . |
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Points observed from the graph |
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1) The graph of sinx and cosx have no breaks and they lie between y=1 and y = -1. So sinx and cosx are continuous for all values of x and the values of sinx and cosx always lie between 1 and +1. Hence sinx and cosx are bounded function. |
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2) There are breaks in the graph of tanx. At these points tan q is not defined and tanx is a discontinuous function. The graph of tanx is not bounded and can assume all real values. |
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Summary |
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1. Trigonometry is that branch of mathematics which deals with the measurements of sides and angles of triangles. |
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2. The figure obtained by rotating a given ray about its end point is called an angle. |
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3. An angle is called negative if the direction of rotation of ray from initial side to terminal. |
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