Use these Inter 1st Year Maths 1A Formulas PDF Chapter 8 Inverse Trigonometric Functions to solve questions creatively.
Intermediate 1st Year Maths 1A Inverse Trigonometric Functions Formulas
→ If sin θ = x and θ ∈ [latex]\left[\frac{-\pi}{2}, \frac{\pi}{2}\right][/latex], then sin-1(x) = θ,
→ If cos θ x and θ ∈ [0, π], then cos-1(x) = θ
→ If tan θ = x and θ ∈ [latex]\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)[/latex]. then tan-1 (y) = θ
→ If cot θ = x and θ ∈ (0, π), then cot-1(x) = θ.
→ If sec θ = x and θ ∈ [o, [latex]\frac{\pi}{2}[/latex]) ∪ ([latex]\frac{\pi}{2}[/latex], π] then sec-1x = θ.
→ If cosec θ = x and θ ∈ [-[latex]\frac{\pi}{2}[/latex], 0) ∪ (o, [latex]\frac{\pi}{2}[/latex]] then cosec-1x = θ.
→ If x ∈ [-1, 1] – {0}, then sin-1(x) = cosec-1[latex]\left(\frac{1}{x}\right)[/latex]
→ If x ∈ [-1, 1] – {0}, then cos-1(x) = sec-1[latex]\left(\frac{1}{x}\right)[/latex]
→ If x > 0, then tan-1(x) = cot-1[latex]\left(\frac{1}{x}\right)[/latex] and
x < 0, then tan-1(x) = cot-1[latex]\left(\frac{1}{x}\right)[/latex] – π
→ sin-1 x + cos-1x = [latex]\frac{\pi}{2}[/latex] (|x| ≤ 1) i.e., – 1 ≤ x ≤ 1
→ tan-1x + cot-1x = [latex]\frac{\pi}{2}[/latex], for any x ∈ R
→ sec-1x + cosec-1x = [latex]\frac{\pi}{2}[/latex], if (-∞, – 1] ∪ [1, ∞)
| Function y = f(x) | Domain (x) | Range (y) |
| (i) sinh-1 (x) | [-1, 1] | [latex] \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] [/latex] |
| (ii) cosh-1 (x) | [1, 1] | [0, π] |
| (iii) tanh-1 (x) | R | [latex] \left(\frac{-\pi}{2}, \frac{\pi}{2}\right) [/latex] |
| (iv) cot-1 (x) | R | (0, π) |
| (v) sec-1 (x) | (-∞, -1] ∪ [1, ∞) | [0, [latex] \frac{\pi}{2} [/latex]) ∪ ([latex] \frac{\pi}{2} [/latex], π] |
| (vi) cosec-1 (x) | (-∞, -1] ∪ [1, ∞) | [-[latex] \frac{\pi}{2} [/latex], 0) ∪ (0, [latex] \frac{\pi}{2} [/latex], π] |
→ Principal values:
For sin-1x, tan-1x, cot-1x, cosec-1x, principal values lies between -[latex]\frac{\pi}{2}[/latex] and [latex]\frac{\pi}{2}[/latex]
For cos-1 x, sec-1 x, principal values lies between 0 and π.
→ tan-1x + tan-1y = tan-1[latex]\left(\frac{x+y}{1-x y}\right)[/latex] if (xy < 1), x > 0, y > 0
= [latex]\frac{\pi}{2}[/latex] if (xy = 1), x > 0, y > 0
= π +tan-1[latex\left(\frac{x+y}{1-x y}\right)][/latex], if (xy > 1), x > 0, y > 0
→ If x < 0, y < 0 then tan-1x + tan-1y = tan-1[latex]\left(\frac{x+y}{1-x y}\right)[/latex] if xy > 1
= – [latex]\frac{\pi}{2}[/latex], if xy < 1
= -[latex]\frac{\pi}{2}[/latex], if xy = 1
→ If x, y ∈ [0, 1] and x2 + y2 ≤ 1, then sin-1 x + sin-1 y = sin-1[latex]\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)[/latex]
→ If x, y ∈ [0, 1] and x2 + y2 > 1 then sin-1 x + sin-1 (y) = π – sin-1[latex]\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)[/latex]
→ If x, y ∈ [0, 1], then sin-1 x + sin-1y) = cos-1([latex]\sqrt{1-x^{2}} \sqrt{1-y^{2}}[/latex] – xy)
→ If x, y ∈ [0, 1] then sin-1 y = sin-1(x[latex]\sqrt{1-y^{2}}[/latex] – y[latex]\sqrt{1-x^{2}}[/latex])
→ If x, y ∈ [0, 1], then cos-1 x + cos-1y = cos-1([latex]\sqrt{1-x^{2}} \sqrt{1-y^{2}}[/latex] + xy)
→ If x, y ∈ [0, 1] then cos-1 x + cos-1y = cos-1(xy – [latex]\sqrt{1-x^{2}} \sqrt{1-y^{2}}[/latex])
→ If x, y ∈ [0, 1] and x2 + y2 ≥ 1 then cos-1x + cos-1y = sin-1(y[latex]\sqrt{1-x^{2}}[/latex] + x[latex]\sqrt{1-y^{2}}[/latex])
→ If 0 ≤ x ≤ y ≤ 1 then cos-1 x – cos-1y = cos-1(xy + [latex]\sqrt{1-x^{2}} \sqrt{1-y^{2}}[/latex])
→ If x, y ∈ [0, 1], then cos-1 x – cos-1y = sin-1(y[latex]\sqrt{1-x^{2}}[/latex] – x[latex]\sqrt{1-y^{2}}[/latex])
→ If x ∈ [-1, 1]- {0}, then sin-1 (x) = cosec-1[latex]\left(\frac{1}{x}\right)[/latex]
→ If x ∈ [-1, 1] – {0}, then cos-1(x) = sec-1[latex]\left(\frac{1}{x}\right)[/latex]
→ If x > 0, then tan-1x = cot-1[latex]\left(\frac{1}{x}\right)[/latex] and
→ If x < 0, then tan-1x = cot-1[latex]\left(\frac{1}{x}\right)[/latex] – π
→ sin-1(-x) = -sin-1(x), if x ∈ [-1, 1]
→ cos-1(-x) = π – cos-1(x), if x ∈ [-1, 1]
→ tan-1(-x) = -tan-1 (x), for any x ∈ R
→ For any x ∈ R, cot-7 (-x) = π – cot-1(x)
→ If x ∈ [-∞, -1] ∪ [1, ∞) then
- sec-1(-x) = π – sec-1(x)
- cosec-1(-x) = -cosec-1(x)
→ If θ ∈ [latex]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right][/latex], then sin-1(sin θ) = θ and if x ∈ [-1, 1], then sin(sin-1x) = x.
→ If θ ∈ [0, π], then cos-1(cos θ) = θ and if x ∈ [-1, 1], then cos (cos-1x) = x.
→ If θ ∈ [latex]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/latex], then tan-1(tan θ) = θ and for any x ∈ R, then tan(tan-1x) = x
→ If θ ∈ (0, [latex]\frac{\pi}{2}[/latex]) ∪ ([latex]\frac{\pi}{2}[/latex], π) then cot-1(cot θ) = θ and for any x ∈ R, cot (cot-1x) = x.
→ If θ ∈ (0, [latex]\frac{\pi}{2}[/latex]) ∪ ([latex]\frac{\pi}{2}[/latex], π) then sec-1(sec θ) = θ and if x ∈ (-∞, -1] ∪ [1, ∞) then sec (sec-1x) = x.
→ If θ ∈ [-[latex]\frac{\pi}{2}[/latex], 0) ∪ (0, [latex]\frac{\pi}{2}[/latex]] then cosec-1 (cosec θ) = θ and if x ∈ (-∞, -1] ∪ [1, ∞), then cosec (cosec-1x) = x.
→ θ ∈ [0, π] sin-1(cos θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ θ ∈ [latex]\left[\frac{\pi}{2}, \frac{\pi}{2}\right][/latex] ⇒ cos-1(sin θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ θ ∈ [latex]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/latex] ⇒ cot-1(tan θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ θ ∈ (0, π) ⇒ tan-1(cot θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ θ ∈ [-[latex]\frac{\pi}{2}[/latex], 0) ∪ (0, [latex]\frac{\pi}{2}[/latex]] ⇒ sec-1(cosec θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ θ ∈ [0, [latex]\frac{\pi}{2}[/latex]) ∪ ([latex]\frac{\pi}{2}[/latex], π] ⇒ cosec-1(sec θ) = [latex]\frac{\pi}{2}[/latex] – θ
→ 0 ≤ x ≤ 1 ⇒ sin-1(x) = cos-1[latex]\left(\sqrt{1-x^{2}}\right)[/latex] and
→ If -1 ≤ x < 0 ⇒ sin-1(x) = -cos-1[latex]\left(\sqrt{1-x^{2}}\right)[/latex]
→ -1 < x < 1 ⇒ sin-1(x) = tan-1[latex]\left(\frac{x}{\sqrt{1-x^{2}}}\right)[/latex]
→ -1 ≤ x < 0 ⇒ cos-1(x) = π – sin-1[latex]\left(\sqrt{1-x^{2}}\right)[/latex] = π + tan-1[latex]\left(\frac{\sqrt{1-x^{2}}}{x}\right)[/latex]
→ 0 ≤ x ≤ 7 ⇒ cos-1(x) = sin-1[latex]\left(\sqrt{1-x^{2}}\right)[/latex] = tan-1[latex]\left(\frac{\sqrt{1-x^{2}}}{x}\right)[/latex]
→ x > 0 ⇒ tan-1(x) = sin-1[latex]\left(\frac{x}{\sqrt{1+x^{2}}}\right)[/latex] = cos-1[latex]\left(\frac{1}{\sqrt{1+x^{2}}}\right)[/latex]
→ tan-1(x) + tan-1(y) + tan-1(z) = tan-1[latex]\left[\frac{x+y+z-x y z}{1-x y-y z-z x}\right][/latex], ifx, y, z have the same sign and xy + yz + zx < 1.
→ 2 sin-1(x) = sin-12x[latex]\sqrt{1-x^{2}}[/latex], if x ≤ [latex]\frac{1}{\sqrt{2}}[/latex]
= π – sin-12x[latex]\sqrt{1-x^{2}}[/latex], if x > [latex]\frac{1}{\sqrt{2}}[/latex]
→ 2 cos-1(x) = cos-1(2x2 – 1), if x ≥ [latex]\frac{1}{\sqrt{2}}[/latex]
= π – cos-1(1 – 2x2), if x < [latex]\frac{1}{\sqrt{2}}[/latex]
→ 2 tan-1(x) = tan-1[latex]\left(\frac{2 x}{1-x^{2}}\right)[/latex], if |x| < 1
= π – tan-1[latex]\left(\frac{2 x}{1-x^{2}}\right)[/latex], if |x| ≥ 1
→ 2tan-1(x) = sin-1[latex]\left(\frac{2 x}{1+x^{2}}\right)[/latex], ∀ x ∈ R
= cos-1[latex]\left(\frac{1-x^{2}}{1+x^{2}}\right)[/latex], if x ≥ 0
= -cos-1[latex]\left(\frac{1-x^{2}}{1+x^{2}}\right)[/latex], if x < 0
= tan-1[latex]\left(\frac{2 x}{1-x^{2}}\right)[/latex], ∀ x ∈ R
→ 3 sin-1(x) = sin-1(3x – 4x3) for 0 ≤ x < [latex]\frac{1}{2}[/latex]
3 cos-1(x) = cos-1(4x3 – 3x) for [latex]\frac{\sqrt{3}}{2}[/latex] ≤ x < 1
3 tan-1(x) = tan-1[latex]\left\{\frac{3 x-x^{3}}{1-3 x^{2}}\right\}[/latex] for 0 ≤ x < [latex]\frac{1}{\sqrt{3}}[/latex]
→ If sin θ = x, we write θ = sin-1x.
→ sin(sin-1x) = x, sin-1(sin θ) = θ if ‘θ‘ ∈ [latex]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right][/latex]
→ cos(cos-1x)=x, cos-1(cos θ) = θ if θ ∈ [0, n]
→ tan (tan-1x) = x, tan-1(tan θ) = θ if θ ∈ [latex]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/latex]
→ That value of sin-1x lying between – [latex]\frac{\pi}{2}[/latex] and[latex]\frac{\pi}{2}[/latex] is called the principal value of sin-1x.
→ That value of cos-1x lying between 0 and π is called the principal value of cos-1x.
→ That value of tan-1x lying between – [latex]\frac{\pi}{2}[/latex] and [latex]\frac{\pi}{2}[/latex] is called the principal value of tan-1x.
→ If -1 ≤ x ≤ 1, then
- sin-1(- x) = – sin-1x
- cos-1(-x) = π – cos-1x
→ If x ∈ R , then
- tan-1(-x) = -tan-1x
- cot-1(-x) = π – cot-1x
→ If x ≤ -1 or x ≥ 1, then
- cosec-1(-x) = -cosec-1x
- sec-1(-x) = π – sec-1x
→ cosec-1x = sin-1[latex]\frac{1}{x}[/latex] (if x ≠ 0)
→ sec-1x = cos-1[latex]\frac{1}{x}[/latex] (if x ≠ 0)
→ cot-1x = tan-1[latex]\frac{1}{x}[/latex] (if x > 0)
= π + tan-1[latex]\frac{1}{x}[/latex] (if x < 0)
→ sin-1x + cos-1x = π/2 ,
→ tan-1x + cot-1x = π/2,
→ sec-1x + cosec-1x = π/2.
→ If sin-1x + sin-1y = π/2, then x2 + y2 = 1.
→ sin(cos-1x) = [latex]\sqrt{1-x^{2}}[/latex], cos(sin-1x) = [latex]\sqrt{1-x^{2}}[/latex]
→ sin-1 = cos-1[latex]\sqrt{1-x^{2}}[/latex] for 0 ≤ x ≤ 1
= -cos-1[latex]\sqrt{1-x^{2}}[/latex] for – 1 ≤ x ≤ 0