Inverse Trigonometric Function Formula

We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function.

(i)  sin (sin\(^{-1}\) x) = x and sin\(^{-1}\) (sin θ) = θ, provided that - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\) and - 1 ≤ x ≤ 1.

(ii) cos (cos\(^{-1}\) x) = x and cos\(^{-1}\) (cos θ) = θ, provided that 0 ≤ θ ≤ π and - 1 ≤ x ≤ 1.

(iii) tan (tan\(^{-1}\) x) = x and tan\(^{-1}\) (tan θ) = θ, provided that - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and - ∞ < x < ∞.

(iv) csc (csc\(^{-1}\) x) = x and sec\(^{-1}\) (sec θ) = θ, provided that - \(\frac{π}{2}\) ≤ θ < 0 or  0 < θ ≤ \(\frac{π}{2}\)  and - ∞ < x ≤ 1 or -1 ≤ x < ∞.

(v) sec (sec\(^{-1}\) x) = x and sec\(^{-1}\) (sec θ) = θ, provided that 0 ≤ θ ≤ \(\frac{π}{2}\) or \(\frac{π}{2}\) <  θ ≤ π and - ∞ < x ≤ 1 or 1 ≤ x < ∞.

(vi)  cot (cot\(^{-1}\) x) = x and cot\(^{-1}\) (cot θ) = θ, provided that 0 < θ < π and - ∞ < x < ∞.

(vii) The function sin\(^{-1}\) x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of sin\(^{-1}\) x then - \(\frac{π}{2}\) ≤ θ ≤ \(\frac{π}{2}\).

(viii) The function cos\(^{-1}\)  x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of cos\(^{-1}\) x then 0 ≤ θ ≤ π.

(ix) The function tan\(^{-1}\) x is defined for any real value of x i.e., - ∞ < x < ∞; if θ be the principal value of tan\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\).

(x)  The function cot\(^{-1}\) x is defined when - ∞ < x < ∞; if θ be the principal value of cot\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and θ ≠ 0.

(xi) The function sec\(^{-1}\) x is defined when, I x I ≥ 1 ; if θ be the principal value of sec\(^{-1}\) x then 0 ≤ θ ≤ π and θ ≠ \(\frac{π}{2}\).

(xii) The function csc\(^{-1}\) x is defined if I x I ≥ 1; if θ be the principal value of csc\(^{-1}\) x then - \(\frac{π}{2}\) < θ < \(\frac{π}{2}\) and θ ≠ 0.

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

(xiv) cos\(^{-1}\) (-x) = π - cos\(^{-1}\) x

(xv) tan\(^{-1}\) (-x) = - tan\(^{-1}\) x

(xvi) csc\(^{-1}\) (-x) = - csc\(^{-1}\) x

(xvii) sec\(^{-1}\) (-x) = π - sec\(^{-1}\) x

(xviii) cot\(^{-1}\) (-x) = cot\(^{-1}\) x

(xix) In numerical problems principal values of inverse circular functions are generally taken.  

(xx) sin\(^{-1}\) x + cos\(^{-1}\) x = \(\frac{π}{2}\)

(xxi) sec\(^{-1}\) x + csc\(^{-1}\) x = \(\frac{π}{2}\).

(xxii) tan\(^{-1}\) x + cot\(^{-1}\) x = \(\frac{π}{2}\)

(xxiii) sin\(^{-1}\) x + sin\(^{-1}\) y = sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\)  + y\(^{2}\) ≤ 1.

(xxiv) sin\(^{-1}\) x + sin\(^{-1}\) y = π - sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\)  + y\(^{2}\) > 1.

(xxv) sin\(^{-1}\) x - sin\(^{-1}\) y = sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\)  + y\(^{2}\) ≤ 1.

(xxvi) sin\(^{-1}\) x - sin\(^{-1}\) y = π - sin\(^{-1}\) (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\)), if x, y ≥ 0 and x\(^{2}\)  + y\(^{2}\) > 1.

(xxvii) cos\(^{-1}\) x + cos\(^{-1}\) y = cos\(^{-1}\)(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\)  + y\(^{2}\) ≤  1.

(xxviii) cos\(^{-1}\) x + cos\(^{-1}\) y = π - cos\(^{-1}\)(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\)  + y\(^{2}\) >  1.

(xxix) cos\(^{-1}\) x - cos\(^{-1}\) y = cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\)  + y\(^{2}\) ≤  1.

(xxx) cos\(^{-1}\) x - cos\(^{-1}\) y = π - cos\(^{-1}\)(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\)), if x, y > 0 and x\(^{2}\)  + y\(^{2}\) >  1.

(xxxi) tan\(^{-1}\) x + tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)), if x > 0, y > 0 and xy < 1.

 (xxxii) tan\(^{-1}\) x + tan\(^{-1}\) y = π + tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)), if x > 0, y > 0 and xy > 1.

(xxxiii) tan\(^{-1}\) x + tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x + y}{1 - xy}\)) - π, if x < 0, y > 0 and xy > 1.

(xxxiv) tan\(^{-1}\) x + tan\(^{-1}\) y + tan\(^{-1}\) z = tan\(^{-1}\) \(\frac{x + y + z - xyz}{1 - xy - yz - zx}\)

(xxxv) tan\(^{-1}\) x - tan\(^{-1}\) y = tan\(^{-1}\) (\(\frac{x - y}{1 + xy}\))

(xxxvi) 2 sin\(^{-1}\) x = sin\(^{-1}\) (2x\(\sqrt{1 - x^{2}}\))

(xxxvii) 2 cos\(^{-1}\) x = cos\(^{-1}\) (2x\(^{2}\) - 1)

(xxxviii) 2 tan\(^{-1}\) x = tan\(^{-1}\) (\(\frac{2x}{1 - x^{2}}\)) = sin\(^{-1}\) (\(\frac{2x}{1 + x^{2}}\)) = cos\(^{-1}\) (\(\frac{1 - x^{2}}{1 + x^{2}}\))

(xxxix) 3 sin\(^{-1}\) x = sin\(^{-1}\) (3x - 4x\(^{3}\))

(xxxx) 3 cos\(^{-1}\) x = cos\(^{-1}\) (4x\(^{3}\) - 3x)

(xxxxi) 3 tan\(^{-1}\) x = tan\(^{-1}\) (\(\frac{3x - x^{3}}{1 - 3x^{2}}\))

 Inverse Trigonometric Functions




11 and 12 Grade Math

From Inverse Trigonometric Function Formula to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 06, 23 01:23 AM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More

  2. Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

    Dec 04, 23 02:14 PM

    Different types of Indian Coins
    Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

    Read More

  3. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Dec 04, 23 01:50 PM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More