Basic Inverse Trigonometric Functions

Q.

The principal value of ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ is

Q. 18.If sin t + cos t = 1/5, then tan t/2 is equal to?

Q. Show tha 4 sin 27 =(5+(5)) - (3-(5))

Q. The fundamental Period of f(x) = sin(cosx) + sin (sinx) is

Q. Q1. The value of sin55^° - sin19^°+ sin53^° - sin17^° is always equal to Q2. Tan9^° - tan27^° - tan63^° +tan81^°=

Q. Write the value of $\sin \frac{\mathrm{\pi }}{15}\sin \frac{4\mathrm{\pi }}{15}\sin \frac{3\mathrm{\pi }}{10}$

Q.

Sin5x-Sinx expressed as a product is

1. 2cos3xsin2x

2. 2cos2xsin3x

3. 2sin3xcos2x

4. 2sin2xcos3x

Q. The range of the functions f(x)= cos^2(x)-5cos(x)-9

Q.

What is the value of sin 35⁰-sin55⁰?

1. 

2. 

3. 1

4. -2sin5⁰

Q. 32. Prove that Sin5-sin67+sin77-sin139+sin149=0

Q. Prove that:
(i) cos 55° + cos 65° + cos 175° = 0

(ii) sin 50° − sin 70° + sin 10° = 0

(iii) cos 80° + cos 40° − cos 20° = 0

(iv) cos 20° + cos 100° + cos 140° = 0

(v) $\sin \frac{5\mathrm{\pi }}{18}-\cos \frac{4\mathrm{\pi }}{9}=\sqrt{3}\sin \frac{\mathrm{\pi }}{9}$

(vi) $\cos \frac{\mathrm{\pi }}{12}-\sin \frac{\mathrm{\pi }}{12}=\frac{1}{\sqrt{2}}$

(vii) sin 80° − cos 70° = cos 50°

(viii) sin 51° + cos 81° = cos 21°

Q. This section contains 1 Matrix Match type question, which has 2 Columns (Column I and Column II). Column I has four entries (A), (B), (C) and (D), Column II has four entries (P), (Q), (R) and (S). Match the entries in Column I with the entries in Column II. Each entry in Column I may match with one or more entries in Column II.

इस खण्ड में 1 मैट्रिक्स मिलान प्रकार का प्रश्न है, जिसमें 2 कॉलम (कॉलम I तथा कॉलम II) हैं। कॉलम I में चार प्रविष्टियाँ (A), (B), (C) तथा (D) हैं, कॉलम II में चार प्रविष्टियाँ (P), (Q), (R) तथा (S) हैं। कॉलम I में दी गयी प्रविष्टियां का मिलान कॉलम II में दी गयी प्रविष्टियों के साथ कीजिए। कॉलम I में दी गयी प्रत्येक प्रविष्टि का मिलान कॉलम-II में दी गयी एक या अधिक प्रविष्टियों के साथ हो सकता है।

Match the Columns and choose the appropriate answer

कॉलमों का मिलान करें और सही उत्तर चुनें
​​​​​​​
Match the functions given in Column-I to their domains given in Column-II .

कॉलम-I में दिये गये फलनों का कॉलम-II में दिये गये इनके प्रांतों के साथ मिलान कीजिए।

	Column-I कॉलम-I		Column-II कॉलम-II
(A)	sin–1(2x)	(P)	R
(B)	tan–1(2x)	(Q)	(1, ∞)
(C)	cos–1(x2 + 1)	(R)	$\left[-\frac{1}{2},\frac{1}{2}\right]$
(D)	loge(x – 1)	(S)	{0}

1. A-S, B-P, C-S, D-P
2. A-Q, B-S, C-R, D-P
3. A-R, B-P, C-S, D-Q
4. A-P, B-R, C-S, D-Q

Q. Domain of the function $f\left(x\right)={\tan }^{-1}\left(\frac{x+4}{(x+5)\sqrt{x^2-4}}\right)$ is

फलन $f\left(x\right)={\tan }^{-1}\left(\frac{x+4}{(x+5)\sqrt{x^2-4}}\right)$ का प्रांत है

1. R – {–2, 2}
2. R – {5}
3. R – [–2, 2]
4. $R-\left([-2,2]\cup \left\{5\right\}\right)$

Q. If $\frac{\tan 2^{\circ }\cdot \tan {2017}^{\circ }\cdot \tan {2019}^{\circ }}{\tan {2019}^{\circ }-\tan {2017}^{\circ }-\tan 2^{\circ }}=a$, then the value of ${\tan }^{-1}a$(in degrees) is

Q. On simplifying $\frac{{\sin }^{3}A+\sin 3A}{\sin A}+\frac{{\cos }^{3}A-\cos 3A}{\cos A}$ we get

1. $\sin 3A$
2. $\cos 3A$
3. $3$
4. $\sin A+\cos A$

Q. find the range of f(x)= 2sin^8 x-3sin^4 x+2

Q. If $cos\left(2si{n}^{-1}x\right)=\frac{1}{9}$, then x=

1. $Only\frac{2}{3}$
2. $Only\frac{-2}{3}$
3. $\frac{2}{3},\frac{-2}{3}$
4. $Neither\frac{2}{3}nor\frac{-2}{3}$

Q. find value of cos 53 degrees and sin 53 degrees

Q. the range of (x^2-x+1/x^2+x+1) is

Q.

$sin\left(2si{n}^{-1}\sqrt{\frac{63}{65}}\right)=$

1. $\frac{2\sqrt{126}}{65}$
   

2. $\frac{4\sqrt{63}}{65}$
   

3. $\frac{8\sqrt{63}}{65}$
   

4. $\frac{\sqrt{65}}{63}$
   

Q. The number of solutions for the equation $co{s}^{-1}(1-x)+mco{s}^{-1}x=\frac{n\pi }{2},wherem>0,n\le 0$ is

1. \N
2. 1
3. 2
4. Infinite

Q. $co{s}^{-1}x=ta{n}^{-1}\frac{\sqrt{1-x^2}}{x}$, then:

1. $X\in R$
2. $-\le x\le 1,x\ne 0$
3. $0<x\le 1$
4. $Noneofthese$

Q. Find the value of the expression
$3{\sec }^{-1}\left(2\right)-6{\sin }^{-1}\left(\frac{1}{2}\right).$

1. $2\pi$
2. $\frac{5\pi }{6}$
3. $0$
4. $1$

Q. sin² π/18 + sin² π/9 + sin² 7π/18 + sin² 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0

Q.

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}$ (cosec x - cot x)

Q. Prove that:
$\sin 6°\sin 42°\sin 66°\sin 78°=\frac{1}{16}$

Q. Value of sin(90+theta)= sin(90-theta)= cos(90+theta)= cos(90-theta)=

Q. Write the value of sin 12° sin 48° sin 54°.

Q. If ${\cos }^{-1}\left(\frac{2}{3x}\right)+{\cos }^{-1}\left(\frac{3}{4x}\right)=\frac{\pi }{2}(x>\frac{3}{4}),$ then $x$ is equal to:

1. $\frac{\sqrt{145}}{12}$
2. $\frac{\sqrt{146}}{12}$
3. $\frac{\sqrt{145}}{11}$
4. $\frac{\sqrt{145}}{10}$

Q. If $2ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}$, then:

1. $x\in R$
2. $x\ge 1$
3. $x\le -1$
4. $-1\le x\le 1$