Trigo Identities

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Trigonometry Identities

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  • TRIGONOMETRIC RATIOS & IDENTITIES

    1. INTRODUCTION

    The word 'Trigonometry' is derived from two Greek words

    (1) Trigonon and(2) MetronThe word trigonon means a triangle and the word metron means a measurement. Hence trigonometry means thescience of measuring triangles.

    2. ANGLE

    Consider a ray OAuuur

    . If this ray rotates about its end point O and takes the position OB , then the angle AOB has

    been generated.

    Vertex O

    = angle

    Initial side A

    B

    Termin

    al side

    An angle is considered as the figure obtained by rotating a given ray about its end - point.

    The initial position OA is called the initial side and the final position OB is called terminal side of the angle. The endpoint O about which the ray rotates is called the vertex of the angle.

    3. SENSE OF AN ANGLE

    The sense of an angle is said to be positive or negative according as the initial side rotates in anticlockwise orclockwise direction to get to the terminal side.

    O = +ve

    Anticlockwise directionA

    B O = ve

    Clockwise directionA

    B

    4. RIGHT ANGLE

    When two lines intersect at a point in such a way that two adjacent angles made by them are equal, then eachangle is called a right angle.

    9090

    OX' X

    A

    5. A CONSTANT NUMBER pipipipipiThe ratio of the circumference to the diameter of a circle is always equal to a constant and this constant is denotedby the Greek letter pi

    TRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATRIGONOMETRIC RATIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIESTIOS & IDENTITIES

  • TRIGONOMETRIC RATIOS & IDENTITIES

    i.e. Circumference of a circle

    Diameter of the circle= pi

    (constant)

    The constant pi is an irrational number and its approximate value is taken as 227 . The more accurate value to six

    decimals places is taken as 355113 .

    6. SYSTEMS OF MEASUREMENT OF AN ANGLESThere are three systems for measuring angles

    6.1 Sexagesimal or English system6.2 Centesimal or French system6.3 Circular system

    6.1 Sexagesimal system : The principal unit in this system is degree (). One right angle is divided into 90 equalpart and each part is called one degree (1) . One degree is divided into 60 equal parts and each part is calledone minute. Minute is denoted by (1'). One minute is equally divided into 60 equal parts and each part is calledone second (1").In Mathematical form :

    One right angle = 90 (Read as 90 degrees )1 = 60' (Read as 60 minutes )1' = 60" (Read as 60 seconds )

    Ex.1 40 30' is equal to

    (1) o41

    2

    (2) 81 (3) o81

    2

    (4) None of these

    Sol. We know that , 30' = o1

    2

    ; 40 + o1

    2

    =

    o812

    6.2 Centesimal system : The principal unit in system is grade and is denoted by (g). One right angle is dividedinto 100 equal parts, called grades, and each grade is subdivided into 100 minutes, and each minute into 100seconds.

    In Mathematical form :

    One right angles = 100g (Read as 100 grades)1g = 100' (Read as 100 seconds)1' = 100" (Read as 100 seconds)

    Ex.2 25' is equal to -Sol. 100' is equal to 1g

    so is equal to g g1 125

    100 4

    =

    Relation between Sexagesimal and Centesimal systems :One right angle = 90 (degree system) ..... (1)

  • TRIGONOMETRIC RATIOS & IDENTITIES

    One right angle = 100g (grade system) ..... (2)by (1) and (2)

    90 = 100g or , D G90 100

    =

    then we can say, ; 1 = g10

    9

    , 1g = o9

    10

    Ex.3 80g is equal to

    Sol. We know that 1g = o9

    10

    then, 80g = o9 80

    10

    80g = 72

    6.3 Circular system : In circular system the unit of measurement is radian. One radian, written as 1C, is the measureof an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.Consider a circle of radius r having centre of O. Let A be a point on the circle. Now cut off an arc AB whoselength is equal to the radius r of the circle.

    B O rIc

    rr

    C

    A

    In the adjacent figure OA = OC = arc AC = r = radius of circle, then measurement of AOC is one radianand denoted by 1c. Thus AOC = 1c .

    6.3.1 Some Important conversion

    pi Radian = 180 One radian = o180

    pi 6pi

    Radian = 30

    4pi

    Radian = 45 3pi

    Radian = 60 2pi

    Radian = 90

    23pi

    Radian = 120 34pi

    Radian = 13556pi

    Radian = 150

    76pi

    Radian = 210 54pi

    Radian = 22553pi

    Radian = 300

  • TRIGONOMETRIC RATIOS & IDENTITIES

    Ex.4 240 is equal to

    [1] C4

    3

    p [2] C3

    4

    p [3] '4

    3p [4]

    '34p

    Sol. We know that180 = pi

    240 = C

    x240180

    p

    =

    C43

    p

    Ans. [1]

    Ex.5 The difference between two acute angle of a right angle triangle is 9p . Then the angles in degree are -

    [1] 50, 30 [2] 25, 45 [3] 20, 40 [4] 35, 55Sol. In triangle ABC let C = 90

    So A B =

    9p = 20 .......... (i)

    and sum of all the angles in ABC

    A + B + C = 180

    C = 90

    A + B = 90 ......... (ii)Solving (i) & (ii) A = 55, B = = 35 Ans. [4]

    6.3.2 Relation between systems of measurement of angles

    D G 2C90 100

    = =

    pi

    Ex.6 The length of an arc of a circle of radius 5 cm subtending a central angle measuring 15 is -

    [1] 312p

    cm [2] 712p

    cm [3] 512p [4] None of these

    Sol. Let s be the length of the arc subtending an angle at the centre of a circle of radius r.

    then, = s

    r

    Here, r = 5 cm, and = 15 = C

    15x180

    p

    = C

    12

    p

    = s

    r = 12

    p =

    s

    5

    = 512

    pcm

  • TRIGONOMETRIC RATIOS & IDENTITIES

    7. TRIGONOMETRICAL RATIOS OR FUNCTIONSLet a line OA makes angle with a fixed line OX and AM is perpendicular from A on OX. Then in right-angledtriangle AMO, trigonometrical ratios (functions) with respect to are defined as follows :

    sin = perpendicular(P)hypotenuse(H)

    cos = base(B)

    hypotenuse(H)

    tan = perpendicular (P)

    Base (B)

    cosec = HP . sec =

    HB , cot =

    BP

    Note :

    (i) Since t-ratios are ratio between two sides of a right angled triangle with respect to an angle, so they arereal numbers.(ii) may be acute angle or obtuse angle or right angle.

    8. RELATIONS BETWEEN TRIGONOMETRICAL RATIOS

    (i) 1 1 1cosec , sec ,cotsin cos tan

    = = = (ii)

    sintancos

    =

    (iii) coscotsin

    = (iv) sin

    2 + cos2 = 1

    (v) 1 + tan2 = sec2 (vi) 1 + cot2 = cosec2Ex.7 If cosec A + cot A = 11/2, then tan A is equal to

    [1] 21/12 [2] 15/16 [3] 44/117 [4] 117/43Sol. Cosec A + cot A = 11/2

    1cosecA cot A+ =

    211 ....... (1)

    cosec A cot A = 211 ..... (2)

    (1) (2) = 2 cot A = 11 22 11- = 11722

    = tan A = 44

    117 Ans [3]

    O B

    M

    H P

    Y

    X

    A

  • TRIGONOMETRIC RATIOS & IDENTITIES

    Ex.8cos

    1 tanq

    q- + sin

    1 cotq

    q- is equal to

    [1] sin cos [2] sin + cos [3] tan + cot [4] tan cot

    Sol.cos

    1 tanq

    q- + sin

    1 cotq

    q-

    =

    cos sinsin cos1- 1cos sin

    q qq qq q

    +

    -

    =

    2 2cos sincos sin cos sin

    q qq q q q

    -

    - -

    =

    2 2cos sincos sin

    q qq q

    -

    -

    = cos + sin Ans [2]Ex.9 tan2 sec2 (cot2 cos2 ) equals

    [1] 0 [2] 1 [3] 1 [4] 2Sol. tan2 sec2 (cot2 cos2 )

    = sec2 (tan2 cot2 tan2 cos2 )

    = sec2 2

    22

    sin1 coscos

    q qq

    - = sec2 (1sin2)

    = sec2. cos2 = 1 Ans. [3]

    9. SIGN OF TRIGONOMETRIC RATIOS(i) All ratios sin, cos, tan cot, sec and cosec are positive

    in Ist quadrant.(ii) sin( or cosec) positive in IInd quadrant, rest are negative.(iii) tan( or cot) positive in IIIrd quadrant, rest are negative.(iv) cos( or sec) positive in IVth quadrant, rest are negative.

    Ex.10 The value of sin and tan if cos = 1213

    -

    and lies in the third quadrant is -

    [1] 513- and 5

    12 [2] 5

    12 and 5

    13- [3] 1213- and

    513

    - [4] none of these

    Sol. We have cos2 + sin2 = 1

    sin = 21 cos q -

    IInd Quadrant Ist Quadrant

    IIIrd Quadrant IVth Quadrant

    x-axis

    y-axis

  • TRIGONOMETRIC RATIOS & IDENTITIES

    In the third quadrant sin is negaitve, therefore

    sin = 21 cos q- sin = 2121

    13

    - = 5

    13

    then, tan = sincos

    qq tan =

    5 13x

    13 12-

    -

    =

    512 Ans.[1]

    Ex.11 If 2p

    < < pi, then 1 sin1 sin

    qq

    -

    + +

    1 sin1 sin

    qq

    +

    -

    is equal to

    [1] 2 cosec [2] 2 cosec [3] 2 sec [4] 2 sec

    Sol. Exp. = 2(1 sin ) (1 sin )

    1 sinq q

    q

    - + +

    - =

    2cosq

    -

    = 2 sec Ans.[4]

    10. DOMAIN AND RANGE OF A TRIGONOMETRICAL FUNCTIONIf f : X Y is a function, defined on the set X, then the domain of the function f, written as Domain is theset of all independent variables x, for which the image f(x) is well defined element of Y, called the co-domainof f.Range of f : X Y is the set of all images f(x) which belongs to Y , i.e.Range f = {f(x) Y:x X} Y The domain and range of trigonometrical functions are tabu