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Inverse Trig Functions

Inverse Trig Functions

The six functions sin-1, cos-1, tan-1, csc-1, sec-1, and cot-1. These are also written arcsin, arccos, arctan, arccsc, arcsec, and arccot.

 

Technical note: Since none of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent are one-to-one, their inverses are not functions. Each trig function can have its domain restricted, however, in order to make its inverse a function. Some mathematicians write these restricted trig functions and their inverses with an initial capital letter (e.g. Cos or Cos-1). However, most mathematicians do not follow this practice. This website does not distinguish between capitalized and uncapitalized trig functions.

 

 

See also

Inverse trigonometry

Key Formula

If y=sin⁡(x), then x=arcsin⁡(y)\text{If } y = \sin(x), \text{ then } x = \arcsin(y) If y=cos⁡(x), then x=arccos⁡(y)\text{If } y = \cos(x), \text{ then } x = \arccos(y) If y=tan⁡(x), then x=arctan⁡(y)\text{If } y = \tan(x), \text{ then } x = \arctan(y)
Where:
  • xx = The angle (output of the inverse trig function)
  • yy = The ratio (input to the inverse trig function)

Worked Example

Problem: Find the exact value of arcsin(√3/2).
Step 1: Recall that arcsin asks: what angle θ has sin(θ) equal to the given value? We need sin(θ) = √3/2.
sin⁡(θ)=32\sin(\theta) = \frac{\sqrt{3}}{2}
Step 2: Remember that the range of arcsin is restricted to [−π/2, π/2] (i.e., −90° to 90°). So the answer must fall in this interval.
−π2≤θ≤π2-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}
Step 3: From the unit circle, sin(π/3) = √3/2, and π/3 (which is 60°) lies within the required range.
sin⁡ ⁣(π3)=32\sin\!\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
Answer: arcsin(√3/2) = π/3 (or 60°)

Another Example

Problem: Find the exact value of arctan(−1).
Step 1: Arctan asks: what angle θ has tan(θ) = −1?
tan⁡(θ)=−1\tan(\theta) = -1
Step 2: The range of arctan is (−π/2, π/2), so the answer must be in this open interval.
−π2<θ<π2-\frac{\pi}{2} < \theta < \frac{\pi}{2}
Step 3: Since tan(−π/4) = −1 and −π/4 is within the allowed range, this is our answer.
tan⁡ ⁣(−π4)=−1\tan\!\left(-\frac{\pi}{4}\right) = -1
Answer: arctan(−1) = −π/4 (or −45°)

Frequently Asked Questions

Does sin⁻¹(x) mean 1/sin(x)?
No. The notation sin⁻¹(x) means the inverse sine function (arcsin), not the reciprocal. The reciprocal 1/sin(x) is written as csc(x). This is a common source of confusion because in most other math contexts, a superscript of −1 does mean a reciprocal.
Why do inverse trig functions have restricted ranges?
Because the original trig functions are periodic — they repeat the same outputs over and over — they are not one-to-one. To define a proper inverse (where each input gives exactly one output), we restrict the original function's domain to an interval where it is one-to-one. This restriction determines the range of the inverse function. For example, arcsin is restricted to output angles in [−π/2, π/2], and arccos outputs angles in [0, π].

Trig Functions vs. Inverse Trig Functions

Trig functions take an angle as input and return a ratio (a number). Inverse trig functions do the reverse: they take a ratio as input and return an angle. For example, sin(30°) = 0.5 goes forward from angle to ratio, while arcsin(0.5) = 30° goes backward from ratio to angle. Trig functions have unrestricted domains (for sine and cosine, all real numbers), but inverse trig functions have limited domains and carefully restricted ranges to ensure they remain proper functions.

Why It Matters

Inverse trig functions are essential whenever you need to find an unknown angle — in triangle problems, physics, engineering, and navigation. They appear throughout calculus, where the derivatives and integrals of inverse trig functions produce important results (for instance, the integral of 1/(1 + x²) is arctan(x) + C). Any time a problem says "find the angle," you are almost certainly using an inverse trig function.

Common Mistakes

Mistake: Confusing sin⁻¹(x) with (sin x)⁻¹ = 1/sin x.
Correction: sin⁻¹(x) means arcsin(x), the inverse function — not the reciprocal. The reciprocal of sin(x) is csc(x). Using the "arc" notation (arcsin, arccos, arctan) can help avoid this confusion.
Mistake: Ignoring the restricted range and giving extra solutions.
Correction: Each inverse trig function has a specific principal range. For example, arcsin always returns a value in [−π/2, π/2]. If you write arcsin(1/2) = 5π/6, that is incorrect because 5π/6 is outside the range, even though sin(5π/6) = 1/2. Always check that your answer falls within the correct range.

Related Terms