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Trignometry Formulae : Quantitative Aptitude Test


  •  Introduction :
    The word 'Trignometry' is derived from the Greek roots ---'tri' meaning 'three' meaning 'an angle'; and 'metron' meaning 'measure'. Thus 'Trignometry' means three angle measure. It is an analytial study of a three angled geometric figure-namely the triangle.

    Some Important rules:

    • A line is a path joining a set of points in a plane.
    • A portion of a line together with its dividing point, is known as a ray.
    • A join or intersection of two rays having a common end point gives an angle. The common end point of an angle is called vertex of the angle. The rays that form the angle are called the sides of the angle.
    • 90° is one measure of a Right Angle.
    • Any angle measuring between 0° and 90° is called an acute angle.
    • Any angle whose measure exceeds 90° but falls below 180° is called an abtuse angle.
    • An angle between 180° and 360° is called reflex angle.
    • When the rotating ray stops after making an angle of rotation ϑ, the place where the rotating ray stops is called the terminal side.
    • If the ray has completed on full revolution about the vertex, then the angle turned is 360°
    Meansurement of angles:
    For mesurement of angles, there are three known systems. They are
    Sexagesimal measurement (British System):
    In this system 1/360 of a turn is called a degree(°), 1/60 of a degree is called a minute(') and 1/60 of a minute is called a second("). 
    Centesimal measurement (French system):
    this system 1/400 of a turn is called a grade(g), 1/100 of a grade is called a minute(') and 1/100 of a minute is called a second ("). 
    Right angle = 100g, 1g = 100'; 1' = 100".
    Note:-
    The minute and seconds in the Sexagesime system are different with the minutes and seconds respectively in the Centesima system.
    Circular measurement (Radian measure):
    The angle subtended by an are of a circle whose length is equal to the radius of the circlr at the center of the circle is called a radian. In this system the unit of measurement is radian(c). 
    1 right angle = π/2 radians(πC/2)
                                                             
    Trigonometric Ratios:
    Let ABC be a right angle triangle. Then with reference to the angle A, we have the following. 
    Sin A = oppositeside/hypotenuse = BC/AC 
    Cos A = adjacentside/hypotenuse = AB/AC 
    Tan A = oppositeside/adjacentside = BC/AB
     

    Sign of the trigonometric ratios in quadrants:-

    1. All trigonometric ratios are positive in I quadrant.
    2. Sine and cosec are positive in II quadrant.
    3. Tan and cot are positive in III quadrant.
    4. Cos and Sec are positive in IV quadrant.
    5. Sexagesimal (English) system
    6. Centesimal (French) system
    7. The radian or circular measure.

    Comeertion of Trigonometric Ratio to quadrants:-

    Angles of the form ( nπ/2 + or - θ)

    1. If n is even Sin Remains as sin : 
      Cos -- cos 
      Tan -- Tan 
      Cosec -- cosec 
      Sec -- sec 
      Cot -- cot

    2. If n is odd Sin changes as cos : 
      Cos -- as Sin 
      Tan -- as Cot 
      Cot -- as Tan 
      Cosec -- as Sec 
      Sec -- as Cosec 

    The values of trigonometric ratios of some standard angles:-

    Some standard formulae:

     
    • Sin θ = 1/cosec θ => sinθ.cosecθ = 1
    • Cosθ = 1/secθ => cosθ.secθ = 1
    • tanθ = 1/ cot => tanθ.cotθ = 1
    • sin2θ+cos2θ = 1 => 1-cos2θ = sin2 and 1 - sin2θ = cos2θ
    • sec2θ - tan2θ = 1 => 1+tan2θ = sec2θ and sec2θ - 1 = tan2θ
    • cosec2θ - cot2θ = 1 => 1 + cot2θ = cosec2θ and cosec2θ - 1 = tan2θ
    • sin2θ = 2sin θ cos θ
    • cos2θ = 2sinθcosθ
    • tan2θ = 2tanθ/1-tan2θ
    • Sin (A+B) = SinA CosB + CosA SinB
    • Sin (A-B) = SinA CosB - CosA SinB
    • Cos (A+B) = CosA CosB - SinA SinB
    • Cos (A-B) = CosA CosB + SinA SinB
    • Tan (A+B) = (tan A + tan B)/(1-tan A tan B)
    • Tan (A-B) = (tan A - tan B)/(1+tan A tan B)
    • Sin 15° = Cos 75° = (√3 - 1)/2√2
    • Cos 15° = Sin 75° = (√3 + 1)/2√2
    • Tan 15° = Cot 75° = (√3 - 1)/(√23 + 1)
    The values of trigonometric ratios of some standard angles:-

    Complementary angles:
     Two angles α,β are said to be the complementary angles if α + β = 90°
    Supplementary angles: Two angles α,β are said to be the supplementary angles if α + β = 180°
     
    • If α,β are the complementary angles then
      1. sinα = cosβ
      2. cosα = sinβ
      3. tanα = cotβ
      4. cotα = tanβ
      5. sin2α + sin2β = 1
      6. cos2α + cos2β = 1
      7. tanα,tanβ = 1
      8. cotα,cotβ = 1
      1. If α,β are the supplementary angles then
         
        1. sinα = sinβ
        2. Sin (A-B) = SinA CosB - CosA SinB
        3. cosα + cosβ = 0
        4. tanα + tanβ = 0
        5. cotα + cotβ = 0
        6. sin2α + cos2β = 0
        7. cos2α + sin2β = 0



        8.  
        9.  
      1. Sides of few right-angled triangles:
         
        1. 3, 4, 5

        2. 8, 15, 17

        3. 9, 40, 41
  • 1. From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is:
A.) 149 m
B.) 156 m
C.) 173 m
D.) 200 m

Answer: Option 'C'

173 m

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