Inverse trigonometric functions undo the basic trig functions by taking a ratio value and returning an angle. They answer the question, what angle gives this sine, cosine, or tangent value. This is important in geometry, physics, engineering, navigation, and any situation where you know side lengths or a slope and need an angle.
Because trig functions repeat, inverse trig functions use restricted ranges so each input gives one principal angle.
The main inverse trig functions are arcsin x, arccos x, and arctan x, also written sin^-1 x, cos^-1 x, and tan^-1 x. For arcsine and arccosine, the input must be between -1 and 1 because sine and cosine ratios cannot go outside that interval. Arctangent accepts every real number because tangent can take any real value.
On a unit circle, inverse trig means starting with a coordinate ratio and locating the angle in the correct restricted range.
Key Facts
- arcsin x returns the angle θ such that sin θ = x and -π/2 ≤ θ ≤ π/2.
- arccos x returns the angle θ such that cos θ = x and 0 ≤ θ ≤ π.
- arctan x returns the angle θ such that tan θ = x and -π/2 < θ < π/2.
- Domain of arcsin x and arccos x: -1 ≤ x ≤ 1.
- Domain of arctan x: all real numbers, written (-∞, ∞).
- If sin θ = opposite/hypotenuse, then θ = arcsin(opposite/hypotenuse).
Vocabulary
- Inverse trigonometric function
- A function that takes a trig ratio as input and returns a corresponding angle as output.
- Principal value
- The single angle chosen by an inverse trig function from its restricted range.
- Restricted range
- A limited set of output angles used so an inverse trig function gives exactly one answer.
- Unit circle
- A circle with radius 1 centered at the origin, used to connect angles with sine, cosine, and tangent values.
- Radian
- A unit of angle measure where 2π radians equals one full turn of 360 degrees.
Common Mistakes to Avoid
- Treating sin^-1 x as 1/sin x is wrong because sin^-1 x means arcsin x in this context, not the reciprocal cosecant.
- Giving all possible angles instead of the principal value is wrong because inverse trig functions must return one angle in the restricted range.
- Using arcsin or arccos with inputs like 1.4 is wrong because sine and cosine ratios must be between -1 and 1.
- Forgetting quadrant restrictions is wrong because arcsin, arccos, and arctan return angles from different ranges, even when related angles have the same reference angle.
Practice Questions
- 1 Evaluate arcsin(1/2) in radians and degrees.
- 2 A right triangle has opposite side 7 and hypotenuse 25 for angle θ. Find θ = arcsin(7/25) to the nearest degree.
- 3 Explain why arccos(-1/2) gives 2π/3, while arcsin(-1/2) gives -π/6, even though both involve the ratio 1/2.