Chapters

  • Chapter 2
  • Relations and Functions
  • 25 questions

Inverse Trigonometric Functions

Definitions, domains, ranges, principal value branches and graphs.

Trigonometric functions repeat, so none of them is one-one on the whole real line and none can be inverted as it stands. The fix is to restrict each function to an interval on which it is one-one — the principal value branch — and invert it there.

Almost every mistake students make in this chapter comes from forgetting that restriction. \sin^{-1}(\sin x) is not always x; it is x only when x already lies inside the principal branch. Getting the domains and ranges genuinely memorised, rather than half-remembered, is what makes the rest of the chapter straightforward.

What you should be able to do

  • State the domain, range and principal value branch of each inverse trigonometric function.
  • Evaluate principal values accurately, including for negative arguments.
  • Sketch and interpret the graphs of the inverse trigonometric functions.
  • Simplify composite expressions such as \sin^{-1}(\sin x) by first checking which branch the argument lies in.

Topics

2 topics

  1. Definition, Domain, Range and Principal Value Branch

    How each trigonometric function is restricted so that it can be inverted, and the principal value that results.

    • Principal value of inverse sine, cosine and tangent
    • Principal value of inverse cosecant, secant and cotangent
    • Composite expressions and branch checking
  2. Graphs of Inverse Trigonometric Functions

    Reading domain, range and monotonicity from the graph of each inverse function.

Formulas

9 items · All formulas

  • Principal value ranges

    \sin^{-1}x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right], \quad \cos^{-1}x \in [0, \pi], \quad \tan^{-1}x \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)

    The three ranges every other result in the chapter depends on.

  • Remaining principal value ranges

    \csc^{-1}x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]\setminus\{0\}, \quad \sec^{-1}x \in [0,\pi]\setminus\left\{\tfrac{\pi}{2}\right\}, \quad \cot^{-1}x \in (0, \pi)

    Note the excluded points: \csc and \sec are undefined where their reciprocals vanish.

  • Negative arguments — symmetric branches

    \sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \csc^{-1}(-x) = -\csc^{-1}x

    Valid because these branches are symmetric about 0, so a negative answer is available.

    Valid when
    x within the appropriate domain.

  • Negative arguments — non-negative branches

    \cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x

    These ranges lie inside [0,\pi], so the answer cannot be negative; it reflects about \frac{\pi}{2} instead.

  • Inverse composed with the function

    \sin\!\left(\sin^{-1}x\right) = x

    Holds for every x in the domain of the inverse function.

    Valid when
    x \in [-1,1] for sine and cosine; x \in \mathbb{R} for tangent.

  • Function composed with the inverse

    \sin^{-1}(\sin x) = x

    The conditional direction: true only when x already lies in the principal branch.

    Valid when
    x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] for sine; x \in [0,\pi] for cosine; x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) for tangent. Otherwise reduce first.

  • Reciprocal relationships

    \csc^{-1}x = \sin^{-1}\!\left(\tfrac{1}{x}\right), \quad \sec^{-1}x = \cos^{-1}\!\left(\tfrac{1}{x}\right), \quad \cot^{-1}x = \tan^{-1}\!\left(\tfrac{1}{x}\right)

    Useful for converting to the three functions whose values you know best.

    Valid when
    |x| \geq 1 for the first two. The cotangent relation holds as written for x > 0; for x < 0, \cot^{-1}x = \pi + \tan^{-1}\frac{1}{x}.

  • Standard principal values

    \sin^{-1}\!\left(\tfrac{1}{2}\right) = \tfrac{\pi}{6}, \quad \sin^{-1}\!\left(\tfrac{1}{\sqrt2}\right) = \tfrac{\pi}{4}, \quad \sin^{-1}\!\left(\tfrac{\sqrt3}{2}\right) = \tfrac{\pi}{3}

    The values that appear most often in examination questions.

  • Values at the boundaries

    \sin^{-1}(0) = 0,\quad \sin^{-1}(1) = \tfrac{\pi}{2},\quad \cos^{-1}(0) = \tfrac{\pi}{2},\quad \cos^{-1}(1) = 0,\quad \cos^{-1}(-1) = \pi

Key points

7 items · All key points

  • \cos^{-1}, \cot^{-1} and \sec^{-1} never return a negative value — their ranges lie inside [0, \pi].

    So an answer of -\frac{\pi}{3} for a \cos^{-1} expression is wrong before you check anything else.

  • \sin^{-1}(\sin x) = x only when x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].

    Outside the branch, find the angle inside the branch with the same sine. \sin^{-1}\!\left(\sin\frac{3\pi}{4}\right) = \frac{\pi}{4}.

  • \sin^{-1}x is not \dfrac{1}{\sin x} — the -1 is not an exponent.

    The reciprocal of \sin x is \csc x, an entirely different function.

  • \sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2} for every x \in [-1,1].

    A frequently useful identity, and a quick check: if you have found both values, they should add to \frac{\pi}{2}.

  • \tan^{-1}x approaches \pm\frac{\pi}{2} but never attains them — the range is an open interval.

  • \sin^{-1}x is undefined for |x| > 1. Watch for questions whose arguments quietly exceed 1.

  • y = \cos^{-1}x and y = \cot^{-1}x are decreasing functions; the other four inverse functions increase.

Common mistakes

6 items · All common mistakes

  • Mistake

    Writing \cos^{-1}(-x) = -\cos^{-1}x, by analogy with the sine rule.

    Instead

    \cos^{-1}(-x) = \pi - \cos^{-1}x.

    Why

    The rule for \sin^{-1} is learnt first and generalised without checking that the cosine branch [0,\pi] contains no negative numbers.

  • Mistake

    Simplifying \sin^{-1}(\sin x) to x regardless of where x lies.

    Instead

    Check the branch first. If x \notin \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], reduce to the angle in the branch with the same sine.

    Why

    The two operations look like inverses that must cancel, and the restriction is easy to forget under examination pressure.

  • Mistake

    Reading \sin^{-1}x as (\sin x)^{-1} = \dfrac{1}{\sin x}.

    Instead

    \sin^{-1} denotes the inverse function. Write \csc x or (\sin x)^{-1} if the reciprocal is meant.

    Why

    The superscript -1 means reciprocal everywhere else in algebra, so the notation genuinely is misleading.

  • Mistake

    Giving a general solution such as n\pi + (-1)^n\frac{\pi}{6} when a principal value was asked for.

    Instead

    A principal value is a single number inside the stated range. General solutions belong to trigonometric equations, not to this chapter.

    Why

    Both topics involve 'find the angle', so the methods get mixed up.

  • Mistake

    Evaluating \sec^{-1} or \csc^{-1} for arguments strictly between -1 and 1.

    Instead

    Their domain is |x| \geq 1. An argument like \sec^{-1}(0.5) is undefined.

    Why

    Students convert to \cos^{-1}(1/x) mechanically without noticing 1/x has left [-1,1].

  • Mistake

    Assuming \tan^{-1}x can equal \frac{\pi}{2} for a very large x.

    Instead

    The range is the open interval \left(-\frac{\pi}{2}, \frac{\pi}{2}\right); \frac{\pi}{2} is an asymptote, never a value.

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