(심화수학) 역삼각함수 (1)

Main ideas / lessons

• Inverse trigonometric functions are defined as the inverses of trigonometric functions, but only after each trig function is restricted so it becomes one-to-one (injective).
• Many trigonometric functions (like (\tan x), (\sin x), (\cos x)) are periodic and not one-to-one over their natural domains, so their inverses do not exist unless you restrict the domain (and correspondingly the range for the inverse).
• Once a function is properly restricted:
  • The inverse graph is obtained by reflecting the original graph across the line (y=x).
  • The inputs/outputs swap roles: what was previously “(x)” becomes “(y)” and vice versa.
• Arc-notation (e.g., (\arcsin), (\arccos), (\arctan)) indicates the inverse relationship and is interpreted as “the angle whose sine/cosine/tangent equals a given value,” within the chosen principal interval.

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Methodology / instructions (how to define each inverse)

General rule (applies to all three)

• Start with a trig function (e.g., (\sin x), (\cos x), (\tan x)).
• Check whether it is one-to-one over its natural domain:
  • If not one-to-one, then restrict the domain of the original function to an interval where it becomes one-to-one.
• With that restriction:
  • The inverse function exists.
  • The inverse’s graph is produced by:
    • Drawing the line (y=x),
    • Reflecting the restricted trig graph across (y=x).
• Interpret the inverse in terms of angles:
  • Example concept: (\arcsin(a)) = the unique angle in the principal interval whose sine equals (a).

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Inverse of sine: (\arcsin x)

Why restriction is needed

• The basic sine description (from the video) indicates the graph is periodic and not one-to-one.
• Because different (x)-values can yield the same (y)-value, the inverse would not be a function unless restricted.

Domain restriction to make it one-to-one

• Restrict the sine function’s “input angle” to:
  • (\left[-\frac{\pi}{2},\ \frac{\pi}{2}\right]) (described as a principal interval in the lecture).
• With this restriction:
  • The function becomes one-to-one.
  • The inverse function also has a one-to-one relationship.

Graph rule

• The inverse graph is the reflection of the restricted sine graph across (y=x).
• The lecture emphasizes:
  • Original graph (black) → reflected inverse graph (red).

Key characteristics / interpretation

• The inverse sine is denoted:
  • (\arcsin x) (video mentions “arc + sine” form).
• Interpretation of “Arc”:
  • It means “the angle” (an arc’s length notion in the video’s explanation).
• Example reasoning used:
  • If asked what angle gives a certain sine value (the example mentioned involves producing (1/2)):
    • Multiple angles satisfy (\sin \theta = 1/2) globally,
    • But the inverse selects the unique angle within the restricted principal interval, giving (\theta=\pi/6) (as stated).

Summary statement (as conveyed)

• To define inverse sine:
  • Restrict sine’s domain to (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]),
  • Then take the inverse and reflect across (y=x).

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Inverse of cosine: (\arccos x)

Why restriction is needed

• (\cos x) is also not one-to-one over all real numbers because it is periodic and repeats values.

Domain restriction

• The lecture states the principal interval for cosine should be:
  • ([0,\ \pi]).
• On this interval, (\cos x) is continuously decreasing, therefore one-to-one.

Graph rule

• The inverse graph is again obtained by reflecting across (y=x).
• The video describes:
  • Original cosine graph (black) from the chosen interval,
  • Reflected red inverse curve.

Range/values swapping concept

• Because it is an inverse:
  • The sign/positive-negative outcomes swap roles relative to the original function.
• Example reasoning used:
  • For (\cos \theta = 1/2):
    • Globally there are multiple solutions,
    • But within ([0,\pi]) there is a unique one,
    • The lecture concludes (\theta=\pi/3).

Summary statement (as conveyed)

• Inverse cosine is defined by:
  • Restricting cosine’s domain to ([0,\pi]),
  • Then reflecting the graph across (y=x),
  • And using (\arccos x) to mean the unique angle in that interval.

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Inverse of tangent: (\arctan x)

Why restriction is needed

• (\tan x) is not one-to-one over its natural domain due to periodicity and repeated outputs across branches.

Domain restriction

• The lecture specifies the principal interval for tangent as an open interval:
  • (\left(-\frac{\pi}{2},\ \frac{\pi}{2}\right)).
• It explains the endpoints are excluded because:
  • At (x=\pm \frac{\pi}{2}), tangent has vertical asymptotes / undefined behavior,
  • Hence the interval is open.

Graph rule and shape

• After restricting to that interval, (\tan x) becomes continuously increasing, hence one-to-one.
• The inverse is obtained via reflection across (y=x).
• The lecture describes that the inverse curve covers the appropriate full range as expected.

Example reasoning used

• Example question described:
  • “How much angle gives tangent inverse for an output of 1?”
• Globally, (\tan \theta = 1) has multiple solutions, but the inverse selects the unique one in the principal interval.
• The lecture concludes:
  • The unique solution in (\left(-\frac{\pi}{2},\frac{\pi}{2}\right)) is (\theta=\pi/4).

Summary statement (as conveyed)

• To define inverse tangent:
  • Restrict (\tan x) to (\left(-\frac{\pi}{2},\frac{\pi}{2}\right)),
  • Then define (\arctan x) as the inverse,
  • Graph is the reflection across (y=x).

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Overall conclusion of the lesson

• The video covers the conditions required to create inverse functions of:
  • (\sin x) → (\arcsin x),
  • (\cos x) → (\arccos x),
  • (\tan x) → (\arctan x).
• Core requirements are:
  • Restrict to a principal interval where the trig function becomes one-to-one,
  • Use graph reflection across (y=x),
  • Interpret arc-trig notation as the unique angle within the chosen interval.

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Speakers / sources featured

• No specific external sources are mentioned.
• Only the video lecturer/speaker (unnamed) is featured.