Find the exact value of cos ⁡ − 1 ( 3 2 ) \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) cos − 1 ( 2 3 <path d="M 95 702 c -2.7 0 -7.17 -2.7 -13.5 -8 c -5.8 -5.3 -9.5 -10 -9.5 -14 c 0 -2 0.3 -3.3 1 -4 c 1.3 -2.7 23.83 -20.7 67.5 -54 c 44.2 -33.3 65.8 -50.3 66.5 -51 c 1.3 -1.3 3 -2 5 -2 c 4.7 0 8.7 3.3 12 10 s 173 378 173 378 c 0.7 0 35.3 -71 104 -213 c 68.7 -142 137.5 -285 206.5 -429 c 69 -144 104.5 -217.7 106.5 -221 c 5.3 -9.3 12 -14 20 -14 H 400000 v 40 H 845.272 s -225.272 467 -225.272 467 s -235 486 -235 486 c -2.7 4.7 -9 7 -19 7 c -6 0 -10 -1 -12 -3 s -194 -422 -194 -422 s -65 47 -65 47 Z M 834 80 H 400000 v 40 H 845 Z" /> )

− π 3 -\frac{\pi}{3} − 3 π

− π 6 -\frac{\pi}{6} − 6 π

Find the exact value of sec ⁡ − 1 ( − 2 ) \sec^{-1}\left(-2\right) sec − 1 ( − 2 )

− π 3 -\frac{\pi}{3} − 3 π

− 2 π 3 -\frac{2\pi}{3} − 3 2 π

cos ⁡ ( sin ⁡ − 1 1 4 + sec ⁡ − 1 4 3 ) \cos\left(\sin^{-1}\frac{\text{1}}{\text{4}}+\sec^{-1}\frac{4}{3}\right) cos ( sin − 1 4 1 + sec − 1 3 4 )

( 3 15 − 7 ) 16 \frac{\left(3\sqrt{15}-\sqrt{7}\right)}{16} 1 6 ( 3 1 5 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> )

( 3 15 − 7 ) 4 \frac{\left(3\sqrt{15}-\sqrt{7}\right)}{4} 4 ( 3 1 5 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> )

( 3 15 + 7 ) 16 \frac{\left(3\sqrt{15}+\sqrt{7}\right)}{16} 1 6 ( 3 1 5 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> + 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> )

( 3 15 + 7 ) 4 \frac{\left(3\sqrt{15}+\sqrt{7}\right)}{4} 4 ( 3 1 5 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> + 7 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> )

Range of function tan ⁡ − 1 x i s \tan^{-1}x\ is tan − 1 x i s

( − π 2 , π 2 ) \left(-\frac{\pi}{2},\frac{\pi}{2}\right) ( − 2 π , 2 π )

R − { π 2 } R-\left\{\frac{\pi}{2}\right\} R − { 2 π }

[ − π 2 , π 2 ] \left[-\frac{\pi}{2},\frac{\pi}{2}\right] [ − 2 π , 2 π ]

cot ⁡ − 1 [ cot ⁡ ( − 9 π 4 ) ] \cot^{-1}\left[\cot\left(\frac{-9\pi}{4}\right)\right] cot − 1 [ cot ( 4 − 9 π ) ]

− π 4 \frac{-\pi}{4} 4 − π

− 3 π 4 \frac{-3\pi}{4} 4 − 3 π

If xy<1, then tan ⁡ − 1 x + tan ⁡ − 1 y = − − − − − − \tan^{-1}x+\tan^{-1}y=------ tan − 1 x + tan − 1 y = − − − − − −

tan ⁡ − 1 ( x + y 1 − x y ) \tan^{-1}\left(\frac{x+y}{1-xy}\right) tan − 1 ( 1 − x y x + y )

tan ⁡ − 1 ( x − y 1 + x y ) \tan^{-1}\left(\frac{x-y}{1+xy}\right) tan − 1 ( 1 + x y x − y )

π + tan ⁡ − 1 ( x + y 1 − x y ) \pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right) π + tan − 1 ( 1 − x y x + y )

π − tan ⁡ − 1 ( x + y 1 − x y ) \pi-\tan^{-1}\left(\frac{x+y}{1-xy}\right) π − tan − 1 ( 1 − x y x + y )

tan ⁡ − 1 x + cot ⁡ − 1 x = π / 2 w h e n \tan^{-1}x+\cot^{-1}x=π/2\ when tan − 1 x + cot − 1 x = π / 2 w h e n

x ∈ [ − π 2 , π 2 ] x∈\left[-\frac{\pi}{2},\frac{\pi}{2}\right] x ∈ [ − 2 π , 2 π ]

x ∈ ( − π 2 , π 2 ) x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right) x ∈ ( − 2 π , 2 π )

Identify the domain of f ( x ) = cos ⁡ − 1 ( 2 x ) f\left(x\right)=\cos^{-1}\left(2x\right) f ( x ) = cos − 1 ( 2 x )

− 1 2 ≤ x ≤ 1 2 -\frac{1}{2}\le x\le\frac{1}{2} − 2 1 ≤ x ≤ 2 1

− 2 ≤ x ≤ 2 -2\le x\le2 − 2 ≤ x ≤ 2

− 1 ≤ x ≤ 1 -1\le x\le1 − 1 ≤ x ≤ 1 − 1 < x < 1 -1<x<1 − 1 < x < 1

− ∞ < x < ∞ -\infty<x<\infty − ∞ < x < ∞

1. The value of cos ⁡ − 1 ( cos ⁡ 7 π 6 ) \cos^{-1\ }\left(\cos\ \frac{7\pi}{6}\right) cos − 1 ( cos 6 7 π ) is

− π 6 -\frac{\pi}{6} − 6 π

sin ⁡ ( sin ⁡ − 1 ( x ) ) = x o n l y i f \sin\left(\sin^{-1\ }\left(x\right)\right)\ =\ x\ only\ if sin ( sin − 1 ( x ) ) = x o n l y i f

∣ x ∣ ≤ 1 \left|x\right|\le1 ∣ x ∣ ≤ 1

∣ x ∣ ≥ 1 \left|x\right|\ge1 ∣ x ∣ ≥ 1

− π 2 ≤ x ≤ π 2 -\frac{\pi}{2}\le x\le\frac{\pi}{2} − 2 π ≤ x ≤ 2 π

0 ≤ x ≤ π 0\le x\le\pi 0 ≤ x ≤ π

inverse Trigonometric Functions

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Quiz

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12th Grade

•

Practice Problem

•

Hard

•

CCSS

HSF.TF.B.7, HSF.TF.C.9, HSF-BF.B.4D

Standards-aligned

SANJEEV KUMAR

Used 238+ times

FREE Resource

21 questions

Show all answers

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

What test will you use to determine if a function is one-to-one?

Veetical Line Test

Horizontal Line Test

One-to-one Test

Function Domain-Range Test

Tags

CCSS.HSF.BF.B.4

CCSS.HSF.IF.C.7

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Find the exact value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

$-\frac{\pi}{3}$

$\frac{\pi}{3}$

$-\frac{\pi}{6}$

$\frac{\pi}{6}$

Tags

CCSS.HSF.TF.B.7

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Find the exact value of $\sec^{-1}\left(-2\right)$

$\frac{2\pi}{3}$

$-\frac{\pi}{3}$

$\frac{4\pi}{3}$

$-\frac{2\pi}{3}$

Tags

CCSS.HSF.TF.B.7

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

What is the restricted domain of the cosine function in order to have an inverse cosine function?

$\left[0,\pi\right]\ -\ \left\{\frac{\pi}{2}\right\}$

$\left[0,\pi\right]$

$\left[-1,1\right]$

$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

Tags

CCSS.HSF-BF.B.4D

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

Inverse Functions

Not Inverse Functions

Tags

CCSS.HSF-BF.B.4B

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

What operation allows us to verify if functions are inverses?

Multiplying

Dividing

Adding

Composing

Tags

CCSS.HSF-BF.B.4B

MULTIPLE CHOICE QUESTION

2 mins • 1 pt

$\tan\left(\sin^{-1}\left(\frac{5}{13}\right)\right)$ is

5/12

5/13

3/13

3/5

Tags

CCSS.HSF.TF.C.8

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inverse Trigonometric Functions

21 questions

What test will you use to determine if a function is one-to-one?

Find the exact value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Find the exact value of $\sec^{-1}\left(-2\right)$

What is the restricted domain of the cosine function in order to have an inverse cosine function?

What operation allows us to verify if functions are inverses?

$\tan\left(\sin^{-1}\left(\frac{5}{13}\right)\right)$ is

$\cos\left(\sin^{-1}\frac{\text{1}}{\text{4}}+\sec^{-1}\frac{4}{3}\right)$

Range of function $\tan^{-1}x\ is$

$\cot^{-1}\left[\cot\left(\frac{-9\pi}{4}\right)\right]$

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Popular Resources on Wayground

Discover more resources for Mathematics

inverse Trigonometric Functions

21 questions

What test will you use to determine if a function is one-to-one?

Find the exact value of cos⁡−1(32)\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)cos−1(23​​)

Find the exact value of sec⁡−1(−2)\sec^{-1}\left(-2\right)sec−1(−2)

What is the restricted domain of the cosine function in order to have an inverse cosine function?

What operation allows us to verify if functions are inverses?

tan⁡(sin⁡−1(513))\tan\left(\sin^{-1}\left(\frac{5}{13}\right)\right)tan(sin−1(135​)) is

cos⁡(sin⁡−114+sec⁡−143)\cos\left(\sin^{-1}\frac{\text{1}}{\text{4}}+\sec^{-1}\frac{4}{3}\right)cos(sin−141​+sec−134​)

Range of function tan⁡−1x is\tan^{-1}x\ istan−1x is

cot⁡−1[cot⁡(−9π4)]\cot^{-1}\left[\cot\left(\frac{-9\pi}{4}\right)\right]cot−1[cot(4−9π​)]

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Popular Resources on Wayground

Discover more resources for Mathematics

Find the exact value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Find the exact value of $\sec^{-1}\left(-2\right)$

$\tan\left(\sin^{-1}\left(\frac{5}{13}\right)\right)$ is

$\cos\left(\sin^{-1}\frac{\text{1}}{\text{4}}+\sec^{-1}\frac{4}{3}\right)$

Range of function $\tan^{-1}x\ is$

$\cot^{-1}\left[\cot\left(\frac{-9\pi}{4}\right)\right]$