IBMathsDPAnalysis & Approaches (AA)HLFlashcards3. Geometry & TrigonometryInverse & Reciprocal Trigonometric Functions

Inverse & Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA))

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  • What are the three reciprocal trigonometric functions?

    The three reciprocal trigonometric functions are:

    • secant, sec theta equals fraction numerator 1 over denominator cos theta end fraction

    • cosecant, cosec theta equals fraction numerator 1 over denominator sin theta end fraction

    • cotangent, cot theta equals fraction numerator 1 over denominator tan theta end fraction

    The identities for sec and cosec are defined in your exam formula booklet.

  • When is sec x undefined?

    Sec x is undefined for values of x for which cos x = 0.

    E.g. plus-or-minus 90 º comma space plus-or-minus 270 º comma space plus-or-minus 450 º comma space...open parentheses plus-or-minus pi over 2 comma space plus-or-minus fraction numerator 3 pi over denominator 2 end fraction comma space plus-or-minus fraction numerator 5 pi over denominator 2 end fraction comma space... close parentheses

  • True or False?

    sin to the power of negative 1 end exponent invisible function application x blank equals fraction numerator 1 over denominator sin space invisible function application x end fraction

    False.

    sin to the power of negative 1 end exponent invisible function application x blank not equal to fraction numerator 1 over denominator sin space invisible function application x end fraction

    It is a common mistake to confuse the inverse trig function with the reciprocal trig function.

  • When is cosec x undefined?

    Cosec x is undefined for values of x for which sin x = 0, i.e. for angles that are multiples of pi (180º).

    E.g. 0 º comma space plus-or-minus 180 º comma space plus-or-minus 360 º comma space... open parentheses 0 comma space plus-or-minus pi comma space plus-or-minus 2 pi comma space... close parentheses

  • True or False?

    cot x is undefined for angles that are multiples of pi.

    True.

    cot x is undefined for angles that are multiples of pi (180º).

    This is because cot x is undefined for all angles where tan x = 0.

  • What is the period of the graph y equals sec space x?

    The period of the graph y equals sec space x is 360º, open parentheses 2 pi space rad close parentheses.

  • What is the range of the graph y equals cot space x?

    The range of the graph of y equals cot space x is y element of straight real numbers.

  • Which reciprocal trig function is shown in the graph below.

    Graph of a reciprocal trig function, showing the curve from -2π to 2π on the x-axis.There are vertical asymptotes at multiples of π and the curve does not cross the x-axis. Local maxima appear at (-π/2, -1) and (3π/2, -1). Local minima appear at (-3π/2, 1) and (π/2, 1).

    The reciprocal trig function shown in the graph is cosec x.

  • What are the two Pythagorean identities for reciprocal trig functions?

    The two Pythagorean identities for reciprocal trig functions are:

    • 1 plus tan squared theta equals sec squared theta

    • 1 plus cot squared theta equals cosec squared theta

    These are both given in the exam formula booklet.

  • What are the three inverse trigonometric functions?

    The three inverse trigonometric functions are:

    • arcsine: arcsin x is the inverse function of sin x

    • arccosine: arc cos x is the inverse function of cos x

    • arctangent: arc tan x is the inverse function of tan x

  • True or False?

    In order for the trig functions sin x, cos x and tan x to have inverse functions, their domains must be restricted.

    True.

    The trig functions sin x, cos x and tan x are all many-to-one functions, so must have their domains restricted in order to have inverse functions.

    • The domain of sin x is restricted to negative pi over 2 less or equal than x less or equal than pi over 2,  negative 90 º less or equal than x less or equal than 90 º.

    • The domain of cos x is restricted to 0 less or equal than x less or equal than pi,  0 less or equal than x less or equal than 180 º.

    • The domain of tan x is restricted to negative pi over 2 less than x less than pi over 2, negative 90 º less or equal than x less or equal than 90 º.

  • What are the domains and ranges of the three inverse trigonometric functions?

    The domains and ranges of the three inverse trigonometric functions are:

    Function

    Domain

    Range

    arcsin x

    negative 1 less or equal than x less or equal than 1

    negative pi over 2 less or equal than arcsin x less or equal than pi over 2

    arc cos x

    negative 1 less or equal than x less or equal than 1

    0 less or equal than arc cos x less or equal than pi

    arctan x

    x element of straight real numbers

    negative pi over 2 less than arc tan x less than pi over 2

  • The graph of which inverse trig function is shown below?

    Graph showing a red curve. The curve starts at (1,0) and moves through (0, π/2) to (-1, π).

    The inverse trig function shown in the graph is arccos x.

  • The graph of which inverse trig function is shown below?

    Graph showing a red curve. The graph has asymptotes at y = -π/2 and y = π/2. The curve starts in the bottom left above the lower asymptote , goes through (0, 0) and finishes at the top right just below the upper asymptotes.

    The inverse trig function shown in the graph is arctan x.

  • The graph of which inverse trig function is shown below?

    Graph showing a red curve. The curve starts at (-1, -π/2) and moves through (0, 0) to (1, π/2).

    The inverse trig function shown in the graph is arcsin x.

  • True or False?

    The symmetries of the trig functions can be used with inverse trig functions when values lie outside of the domain or range.

    True.

    The symmetries of the trig functions can be used when values lie outside of the domain or range.

    E.g. using sin open parentheses x close parentheses equals sin open parentheses pi minus x close parentheses you can get

    arcsin open parentheses sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses equals arcsin open parentheses sin open parentheses straight pi minus fraction numerator 2 straight pi over denominator 3 end fraction close parentheses close parentheses

    arcsin open parentheses sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses equals arcsin open parentheses sin open parentheses straight pi over 3 close parentheses close parentheses equals pi over 3

    Note that arcsin open parentheses sin open parentheses fraction numerator 2 pi over denominator 3 end fraction close parentheses close parentheses not equal to fraction numerator 2 pi over denominator 3 end fraction, because fraction numerator 2 pi over denominator 3 end fraction is not in the range of arcsin x.

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