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JEE Maths Trigonometry

Trigonometry PDF Notes, Important Questions and Synopsis

SYNOPSIS

  1. If the direction of rotation is anticlockwise, then the angle is said to be positive. If the direction of rotation is clockwise, then the angle is negative.


    Positive angle — anticlockwise


    Negative angle - clockwise
  2. The angle subtended at the centre by an arc of length 1 unit in a unit circle is said to have a measure of 1 radian.

  3. Even function: A function f(x) is said to be an even function if begin mathsize 12px style straight f open parentheses negative straight x close parentheses equals straight f open parentheses straight x close parentheses end style for all x in its domain.
  4. Odd function: A function f(x) is said to be an odd function if  begin mathsize 12px style straight f open parentheses negative straight x close parentheses equals negative straight f open parentheses straight x close parentheses end style for all x in its domain.
  5. Cosine is an even function and sine is an odd function.
    cos(−x) = cos x
    sin(−x) = −sin x

  6. Signs of trigonometric functions in various quadrants:

     

    I

    II

    III

    IV

    sin x

    +

    +

    cos x

    +

    +

    tan x

    +

    +

    cosec x

    +

    +

    sec x

    +

    +

    cot x

    +

    +

  7. In quadrants, where the y-axis is positive (i.e. I and II), sine is positive, and in quadrants where the x-axis is positive (i.e. I and IV), cosine is positive.
  8. A simple rule to remember the sign of the trigonometric ratios in all the four quadrants is the phrase-All Silver Tea Cups.
  9. A function ‘f’ is said to be a periodic function if there exists a real number T > 0 such that
    f(x + T) = f(x) for all ‘x’. ‘T’ is the period of the function.

  10. sin(2Π + x) = sin x, so the period of sine is 2Π. Period of its reciprocal is also 2Π.

  11. cos(2Π + x) = cos x, so the period of cosine is 2Π. Period of its reciprocal is also 2Π.

  12. tan(Π + x) = tan x. Period of tangent and cotangent function is Π. 

  13. The graph of cos x can be obtained by shifting the sine function along the x-axis by the factor  begin mathsize 12px style straight pi over 2. end style

  14. The tan function differs from sine and cosine functions in two ways:

    1.  Function tan is not defined at the odd multiples of Π/2. 
    2. The tan function is not bounded.
  15.  

    Function

    Period

    y = sin x

    y = sin (ax)

     begin mathsize 12px style fraction numerator text 2 end text straight pi over denominator text a end text end fraction end style

    y = cos x

    y = cos (ax)

     begin mathsize 12px style fraction numerator text 2 end text straight pi over denominator text a end text end fraction end style

    y = cos 3x

     begin mathsize 12px style fraction numerator text 2 end text straight pi over denominator text 3 end text end fraction end style

    y = sin 5x

     begin mathsize 12px style fraction numerator text 2 end text straight pi over denominator text 5 end text end fraction end style
  16. For a function of the form y = k f(ax + b), the range will be ‘k’ times the range of function x, where k is any real number.

    1. If f(x) = sine function in the above form, the range will be equal to [−k, k].
    2. If f(x) = cosine function in the above form, the range will be equal to R−[−k, k].
    3. If the function is of the form ‘k sec (ax + b)’ or ‘k cosec (ax + b)’, the period is equal to the period of function ‘f’ divided by ‘a’.
    4. The position of the graph of y = k f(ax + b) is ‘b’ units to the right or left of y = f(x) depending on whether b < 0 or b > 0.
  17. The solutions of a trigonometric equation for which 0 ≤ x ≤ 2Π are called principal solutions

  18. The expression involving an integer ‘n’ which gives all the solutions of a trigonometric equation is called the general solution.

  19. The numerically smallest value of the angle (in degree or radian) satisfying a given trigonometric equation is called the principal value. If there are two values, one positive and the other negative, which are numerically equal, then the positive value is taken as the principal value.

  20. Trigonometric equations:

    No.

    Equations

    General solution

    Principal value

    1

    sin θ = 0

    θ = np, nÎZ

    θ = 0

    2

    cos θ = 0

    θ = (2n + 1)p/2, nÎZ

    θ = p/2

    3

    tan θ = 0

    θ = np

    θ = 0

    4

    sin θ = sin α

    θ = np + (–1)ⁿ α, nÎZ

    θ = α

    5

    cos θ = cos α

    θ = 2np ± α, nÎZ

    θ = 2α, α > 0

    6

    tan θ = tan α

    θ = np + α, nÎZ

    θ = α

    7

     begin mathsize 12px style sin squared straight theta equals sin squared straight alpha end style  begin mathsize 12px style straight theta equals nπ plus-or-minus straight alpha comma straight n element of straight Z end style

     

    8

     begin mathsize 12px style cos squared straight theta equals cos squared straight alpha end style  begin mathsize 12px style straight theta equals nπ plus-or-minus straight alpha comma straight n element of straight Z end style

     

    9

     begin mathsize 12px style tan squared straight theta equals tan squared straight alpha end style  begin mathsize 12px style straight theta equals nπ plus-or-minus straight alpha comma straight n element of straight Z end style

     

  21. Sine rule:

    begin mathsize 12px style table attributes columnalign left end attributes row cell straight a over sinA equals straight b over sinB equals straight c over sinC end cell row Or row cell sinA over straight a equals sinB over straight b equals sinC over straight c end cell end table end style
  22. Law of cosine: 

    begin mathsize 12px style table attributes columnalign left end attributes row cell text a end text to the power of text 2 end text end exponent equals straight b squared plus straight c squared minus 2 bccosA comma text             cosA  =   end text fraction numerator text b end text to the power of text 2 end text end exponent plus straight c squared minus straight a squared over denominator 2 bc end fraction end cell row cell text b end text to the power of text 2 end text end exponent equals text    end text straight c squared plus straight a squared minus 2 accosB comma text              cosB  =   end text fraction numerator straight a to the power of text 2 end text end exponent plus straight c squared minus straight b squared over denominator 2 ac end fraction end cell row cell straight c to the power of text 2 end text end exponent equals text    end text straight a squared plus straight b squared minus 2 abcosC comma text            cosC  =   end text fraction numerator straight a to the power of text 2 end text end exponent plus straight b squared minus straight c squared over denominator 2 ab end fraction end cell end table end style
  23. Napier’s analogy (Law of tangents):
    1. begin mathsize 12px style tan open parentheses fraction numerator straight B minus straight C over denominator 2 end fraction close parentheses equals open parentheses fraction numerator straight b minus straight c over denominator straight b plus straight c end fraction close parentheses cot straight A over 2 end style
    2. begin mathsize 12px style tan open parentheses fraction numerator straight A minus straight B over denominator 2 end fraction close parentheses equals open parentheses fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction close parentheses cot straight C over 2 end style
    3. begin mathsize 12px style tan open parentheses fraction numerator straight C minus straight A over denominator 2 end fraction close parentheses equals open parentheses fraction numerator straight c minus straight a over denominator straight c plus straight a end fraction close parentheses cot straight B over 2 end style

  24. Area of a triangle having sides a, b and c is given by
    begin mathsize 12px style straight capital delta text   =  end text fraction numerator text 1 end text over denominator text 2 end text end fraction bcsinA equals fraction numerator text 1 end text over denominator text 2 end text end fraction casinB equals fraction numerator text 1 end text over denominator text 2 end text end fraction absinC end style
  25. Graphs of standard trigonometric functions:
    1. Graph of sin x

    2. Graph of cos x

    3. Graph of tan x

    4. Graph of sec x

    5. Graph of cosec x

    6. Graph of cot x



  26. If f is one-to-one and onto (bijection), then there exists an inverse of a function f denoted by begin mathsize 12px style straight f to the power of negative 1 end exponent end style such that begin mathsize 12px style straight f to the power of negative 1 end exponent open parentheses straight y close parentheses equals straight x end style.
  27. Inverse trigonometric functions map real numbers back to angles.
  28. Graphs of inverse trigonometric functions: