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Inverse Trigonometric function - Notes, Concept and All Important Formula

INVERSE TRIGONOMETRIC FUNCTION

1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC

FUNCTIONS :

(a)

Graph of arc sin(x) or graph of sin^(-1)x

f−1:[−1,1]→[−π/2,π/2]f−1:[−1,1]→[−π/2,π/2],

f−1(x)=sin−1(x)f−1(x)=sin−1(x)

(b)

Graph of arc cos(x) or graph of cos^(-1)x

f−1:[−1,1]→[0,π]f−1:[−1,1]→[0,π],

f−1(x)=cos−1xf−1(x)=cos−1x

(c)

Graph of arc tan(x) or graph of tan^(-1)x

f−1:R→(−π/2,π/2)f−1:R→(−π/2,π/2),

f−1(x)=tan−1xf−1(x)=tan−1x

(d)

Graph of arc cot(x) or graph of cot^(-1)x

f−1:R→(0,π)f−1:R→(0,π)

f−1(x)=cot−1xf−1(x)=cot−1x

(e)

Graph of arc sec(x) or graph of sec^(-1)x

f−1:(−∞,−1]∪[1,∞)f−1:(−∞,−1]∪[1,∞)→[0,π/2)∪(π/2,π]→[0,π/2)∪(π/2,π],

f−1(x)=sec−1x

(f)

Graph of arc cosec(x) or graph of cosec^(-1)x

f−1:(−∞,−1]∪[1,∞)→[−π/2,0)∪(0,π/2]

f−1(x)=cosec−1x


2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS:

Property-1 :

(i) y=sin(sin−1x)=x,x∈[−1,1],y∈[−1,1],y is aperiodic

(ii) y=cos(cos−1x)=x,x∈[−1,1],y∈[−1,1],y is aperiodic

(iii) y=tan(tan−1x)=x,x∈R,y∈R,y is aperiodic

(iv) y=cot(cot−1x)=x,x∈R,y∈R,y is aperiodic

(v) y=cosec(cosec−1x)=x,|x|≥1,|y|≥1,y is aperiodic

(vi) y=sec(sec−1x)=x,|x|≥1;|y|≥1,y is aperiodic

Property 2 :

(i) y=sin−1(sinx),x∈R,y∈[−π2,π2]. Periodic with period 2π.

sin−1(sinx)={−π−x,−3π2≤x≤−π2x,−π2≤x≤π2π−x,π2≤x≤3π2x−2π,3π2≤x≤5π23π−x,5π2≤x≤7π2x−4π,7π2≤x≤9π2

Graph of arc sin(sin x)

(ii) y=cos−1(cosx),x∈R,y∈[0,π], periodic with period 2π

cos−1(cosx)={−x,−π≤x≤0x,0≤x≤π2π−x,π≤x≤2πx−2π,2π≤x≤3π4π−x,3π≤x≤4π

Graph of arc cos(cos x)

(iii) y=tan−1(tanx)  x∈R−{(2n−1)π2,n∈I};y∈(−π2,π2), periodic with period π

tan−1(tanx)={x+π,−3π2<x<−π2x,−π2<x<π2x−π,π2<x<3π2x−2π,3π2<x<5π2x−3π,5π2<x<7π2

Graph of arc tan(tan x)

(iv) y=cot−1(cotx),x∈R−{nπ},y∈(0,π), periodic with period π

Graph of arc cot(cot x)

(v) y=cosec−1(cosecx), x∈R−{nπ,n∈I} y∈[−π2,0)∪(0,π2], y is periodic with period 2π.

Graph of arc cosec(cosec x)

(vi) y=sec−1(secx),y is periodic with period 2π

x∈R−{(2n−1)π2n∈I},y∈[0,π2)∪(π2,π]

Graph of arc sec(sec x)

Property 3 :

(i) cosec−1x=sin−11x;x≤−1 or x≥1

(ii) sec−1x=cos−11x;x≤−1 or x≥1

(iii) cot−1x=tan−11x;x>0

=π+tan−11x;x<0

Property 4:

(i) sin−1(−x)=−sin−1x,−1≤x≤1

(ii) tan−1(−x)=−tan−1x,x∈R

(iii) cos−1(−x)=π−cos−1x,−1≤x≤1

(iv) cot−1(−x)=π−cot−1x,x∈R

(v) sec−1(−x)=π−sec−1x,x≤−1 or x≥1

(vi) cosec−1(−x)=−cosec−1x,x≤−1 or x≥1

Property 5 :

(i) sin−1x+cos−1x=π2,−1≤x≤1

(ii) tan−1x+cot−1x=π2,x∈R

(iii) cosec−1x+sec−1x=π2,|x|≥1

Property 6:

(i) tan−1x+tan−1y={tan−1x+y1−xy, where x>0,y>0&xy<1π+tan−1x+y1−xy, where x>0,y>0&xy>1π2, where x>0,y>0&xy=1

(ii) tan−1x−tan−1y=tan−1x−y1+xy, where x>0, y>0

(iii) sin−1x+sin−1y=sin−1[x√1−y2+y√1−x2]
where x>0,y>0 & (x2+y2)<1

Note that : x2+y2<1⇒0<sin−1x+sin−1y<π2

(iv) sin−1x+sin−1y=π−sin−1[x√1−y2+y√1−x2],
where x>0,y>0 & x2+y2>1

Note that : x2+y2>1⇒π2<sin−1x+sin−1y<π

(v) sin−1x−sin−1y=sin−1[x√1−y2−y√1−x2] where x>0,y>0

(vi) cos−1x+cos−1y=cos−1[xy−√1−x2√1−y2], where x>0,y>0

(vii) cos−1x−cos−1y={cos−1(xy+√1−x2√1−y2);x<y,x,y>0−cos−1(xy+√1−x2√1−y2);x>y,x,y>0

(viii) tan−1x+tan−1y+tan−1z=tan−1[x+y+z−xyz1−xy−yz−zx]
if x>0,y>0,z>0 & xy+yz+zx<1

Note : In the above results x & y are taken positive. In case if these are given as negative, we first apply P−4 and then use above results.


3. SIMPLIFIED INVERSE TRIGONOMETRIC FUNCTIONS :

(a) 

Graph of arc sin(2x/(1+x²))

y=f(x)=sin−1(2x1+x2)

=[2tan−1x if ∣x≤1π−2tan−1x if x>1−(π+2tan−1x) if x<−1

(b) 

Graph of arc cos((1-x²)/(1+x²))

y=f(x)=cos−1(1−x21+x2)

=[2tan−1x if x≥0−2tan−1x if x<0

(c) 

Graph of arc tan(2x/(1-x²))

y=f(x)=tan−12x1−x2

=[2tan−1x if |x|<1π+2tan−1x if x<−1−(π−2tan−1x) if x>1

(d)

Graph of arc sin(3x-4x³)

y=f(x)=sin−1(3x−4x3)

=[−(π+3sin−1x) if −1≤x≤−123sin−1x if −12≤x≤12π−3sin−1x if 12≤x≤1

(e)
Graph of arc cos(4x³-3x)

 y=f(x)=cos−1(4x3−3x)  

={3cos−1x−2π if −1≤x≤−122π−3cos−1x if −12≤x≤123cos−1x if 12≤x≤1

(f) 

Graph of arc sin(2x√(1-x²))

sin−1(2x√1−x2)

={−(π+2sin−1x)−1≤x≤−1√22sin−1x−1√2≤x≤1√2π−2sin−1x1√2≤x≤1

(g)

graph of arc cos(2x²-1)

cos−1(2x2−1)

={2cos−1x0≤x≤12π−2cos−1x−1≤x≤0

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