Inverse trigonometric functions
Principal values and their ranges.
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Inverse trig answers the question "which angle?"
Look: The red line is height ½. It meets the sine curve at 30°.
✏️ Your turn!
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1sin⁻¹(1) = ?
2cos⁻¹(½) = ?
3tan⁻¹(0) = ?
! Careful!
✗sin⁻¹x = 1 ÷ sin x.
✓It means "the angle whose sine is x".
✗Giving any angle that works.
✓Give the principal one (sin⁻¹ between −90° and 90°).
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Inverse trig answers the question "which angle?"
sin 30° = ½, so sin⁻¹(½) = 30°. Many angles share a sine, so each inverse gives one agreed answer (the principal value).
Think of it like thissin tells you the height for an angle. sin⁻¹ works backwards like a detective: given the height, find the angle.
Solved together
Find tan⁻¹(1).
Which angle has tan = 1? tan = opposite ÷ adjacent = 1 means the two sides are equal.
Equal sides make a 45° triangle.
tan 45° = 1
Answertan⁻¹(1) = 45° (π/4).
For older learners and grown-ups: a little more
Principal ranges: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2). So cos⁻¹(−½) = 2π/3, not −π/3.
Now practise it
Kanakku makes new questions on inverse trigonometric functions every time. It shows the working and gives a hint when you are stuck.