JEE Main Previous Year Trigonometry Questions and Solutions

Q 21 :

Let $a \leq \tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x \leq b .$ If $α$ and $β$ denote the minimum and maximum possible values of $a$ and $b$ respectively, then:

$α = 0, β = π$
$α = 0, β = \frac{π}{2}$
$α = \frac{π}{2}, β = π$
$α = - \frac{π}{2}, β = \frac{π}{2}$

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(1)

Ans. $α = 0, β = π$

Explanation:

We know that

$\tan^{- 1} x + \cot^{- 1} x = \frac{π}{2}$

$\tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x = \frac{π}{2} + \sin^{- 1} x$

$\frac{- π}{2} \leq \sin^{- 1} x \leq \frac{π}{2}$

$\Rightarrow - \frac{π}{2} + \frac{π}{2} \leq \sin^{- 1} x + \frac{π}{2} \leq \frac{π}{2} + \frac{π}{2}$

$\Rightarrow 0 \leq \frac{π}{2} + \sin^{- 1} x \leq π$

$\Rightarrow 0 \leq \tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x \leq π$

$∴$ $α = 0, β = π$

Q 22 :

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

$- \frac{2 π}{3}$
$- \frac{π}{3}$
$\frac{4 π}{3}$
$\frac{5 π}{3}$

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(2)

Ans. $- \frac{π}{3}$

Explanation:

$\sin^{- 1} (\frac{- \sqrt{3}}{2}) = \sin^{- 1} (- \sin \frac{π}{3})$

$= - \sin^{- 1} (\sin \frac{π}{3}) = - \frac{π}{3}$

Q 23 :

The domain of the function defined by $f (x) = \sin^{- 1} x + \cos x$ is:

$[- 1, 1]$
$[- 1, π + 1]$
$(- \infty, \infty)$
$ϕ$

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Ans. $[- 1, 1]$

Explanation:

The domain of $\cos x$ is $R$ and the domain of $\sin^{- 1} x$ is $[- 1, 1] .$

$∴$ The domain of $\cos x + \sin^{- 1} x$ is $R \cap [- 1, 1]$ , i.e., $[- 1, 1] .$

Q 24 :

The principal value of $\sec^{- 1} (\frac{- 2}{\sqrt{3}})$ is:

$\frac{5 π}{6}$
$\frac{2 π}{3}$
$\frac{π}{3}$
None of these

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Ans. $\frac{5 π}{6}$

Explanation:

$\sec^{- 1} (\frac{- 2}{\sqrt{3}}) = π - \sec^{- 1} (\frac{2}{\sqrt{3}})$

$= π - \frac{π}{6} = \frac{5 π}{6}$

Q 25 :

If $\sin^{- 1} x = y,$ then:

$0 \leq y \leq x$
$\frac{- π}{2} \leq y \leq \frac{π}{2}$
$0 < y < π$
$\frac{- π}{2} < y < \frac{π}{2}$

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(2)

Ans. $\frac{- π}{2} \leq y \leq \frac{π}{2}$

Explanation:

Range of $\sin^{- 1} x$ is $[\frac{- π}{2}, \frac{π}{2}]$

$∴$ $\frac{- π}{2} \leq y \leq \frac{π}{2}$

Q 26 :

The domain of the function $\cos^{- 1} (2 x - 1)$ is:

$[0, 1]$
$[- 1, 1]$
$(- 1, 1)$
$[0, π]$

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Ans. [0, 1]

Explanation:

We know that $\cos^{- 1} x$ is defined for $x \in [- 1, 1]$

$∴$ $f (x) = \cos^{- 1} (2 x - 1)$ is defined if

$- 1 \leq 2 x - 1 \leq 1$

$\Rightarrow$ $0 \leq 2 x \leq 2$

$\Rightarrow$ $0 \leq x \leq 1$

Q 27 :

The value of $\cos^{- 1} (\cos \frac{3 π}{2})$ is equal to:

$\frac{π}{2}$
$\frac{3 π}{2}$
$\frac{5 π}{2}$
$\frac{7 π}{2}$

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Ans. $\frac{π}{2}$

Explanation:

$\cos^{- 1} (\cos \frac{3 π}{2}) \neq \frac{3 π}{2}$ as $\frac{3 π}{2} \notin [0, π]$

$∴$ $\cos^{- 1} (\cos \frac{3 π}{2}) = \cos^{- 1} (0) = \frac{π}{2}$

Q 28 :

The value of $\sin [2 \sin^{- 1} (0.6)]$ is:

$\cdot 48$
$\cdot 96$
$1 \cdot 2$
$\sin 1 \cdot 2$

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Ans. $\cdot 96$

Explanation:

Let $\sin^{- 1} (\cdot 6) = θ,$ i.e., $\sin θ = \cdot 6$

Now, $\sin [2 \sin^{- 1} (0.6)] = \sin (2 θ) = 2 \sin θ \cos θ$

$= 2 (\cdot 6) (\cdot 8) = \cdot 96$

$(∵ \cos θ = \sqrt{1 - \sin^{2} θ} = \sqrt{1 - {(0.6)}^{2}} = 0.8)$

Q 29 :

If $\cot^{- 1} (- \frac{1}{5}) = θ$ , the value of $\sin θ$ is:

$\frac{\sqrt{26}}{5}$
$- \frac{5}{\sqrt{26}}$
$\frac{\sqrt{5}}{\sqrt{26}}$
$\frac{5}{\sqrt{26}}$

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(4)

Ans. $\frac{5}{\sqrt{26}}$

Explanation:

Given, $\cot^{- 1} (\frac{- 1}{5}) = θ, where θ \in (0, π)$

$∴$ $\cot θ = \frac{- 1}{5}$

$∵$ $\sin θ = \frac{1}{c o s e c θ} = \frac{1}{\sqrt{1 + \cot^{2} θ}}$

$= \frac{1}{\sqrt{1 + {(- \frac{1}{5})}^{2}}} = \frac{1}{\sqrt{1 + \frac{1}{25}}} = \frac{1}{\sqrt{\frac{26}{25}}}$

$= \frac{5}{\sqrt{26}}$

Q 30 :

$\tan^{- 1} \sqrt{3} - \sec^{- 1} (- 2)$ is equal to:

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

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Ans. $- \frac{π}{3}$

Explanation:

$\tan^{- 1} \sqrt{3} - \sec^{- 1} (- 2)$

$= \tan^{- 1} \sqrt{3} - (π - \sec^{- 1} 2)$

$= \frac{π}{3} - π + \sec^{- 1} 2$

$= \frac{π}{3} - π + \frac{π}{3}$

$= \frac{2 π}{3} - π$

$= - \frac{π}{3}$

Q 31 :

The value of $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$ is:

$\frac{3 + \sqrt{5}}{2}$
$\frac{3 - \sqrt{5}}{2}$
$\frac{- 3 + \sqrt{5}}{2}$
$\frac{- 3 - \sqrt{5}}{2}$

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Ans. $\frac{3 - \sqrt{5}}{2}$

Explanation:

Let $y = \tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$

Putting, $x = \cos^{- 1} (\frac{\sqrt{5}}{3})$

$\Rightarrow$ $\cos x = \frac{\sqrt{5}}{3}$ ...(i)

Now, $y = \tan (\frac{x}{2})$

$\Rightarrow$ $y = \sqrt{\frac{1 - \cos x}{1 + \cos x}}$

$\Rightarrow$ $y = \sqrt{\frac{1 - \frac{\sqrt{5}}{3}}{1 + \frac{\sqrt{5}}{3}}}$ [From (i)]

$\Rightarrow$ $y = \sqrt{\frac{3 - \sqrt{5}}{3 + \sqrt{5}}}$ $= \sqrt{\frac{(3 - \sqrt{5}) (3 - \sqrt{5})}{(3 + \sqrt{5}) (3 - \sqrt{5})}}$

$\Rightarrow$ $y = \sqrt{\frac{{(3 - \sqrt{5})}^{2}}{9 - 5}}$ $= \frac{3 - \sqrt{5}}{\sqrt{4}}$

$\Rightarrow$ $y = \frac{3 - \sqrt{5}}{2}$

i.e., $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})] = \frac{3 - \sqrt{5}}{2}$

Q 32 :

Match List I with List II.

List I List II

A. The range of $\sin^{- 1} x$ is I. $[\frac{- π}{2}, \frac{π}{2}] - {0}$

B. The range of $\tan^{- 1} x$ is II. $(\frac{π}{2}, \frac{π}{2})$

C. The range of ${cosec}^{- 1} x$ is III. $[0, π] - {\frac{π}{2}}$

D. The range of $\sec^{- 1} x$ is IV. $[\frac{- π}{2}, \frac{π}{2}]$

Choose the correct answer from the option given below.

A-IV, B-III, C-II, D-I
A-IV, B-I, C-III, D-II
A-I, B-IV, C-II, D-III
A-IV, B-II, C-I, D-III

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Ans. A-IV, B-II, C-I, D-III

Q 33 :

If the equation $x^{3} + b x^{2} + c x + 1 = 0 (b < c)$ has only one real root $α$ . Then the value of $2 \tan^{- 1} (cosec α) + \tan^{- 1} (2 \sin α \sec^{2} α)$ is:

$- \frac{π}{2}$
$- π$
$\frac{π}{2}$
$π$

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(2)

$x^{3} + b x^{2} + c x + 1 = 0$

$f (- 1) = b - c < 0$

$f (0) = 1 > 0$

$B \in (0, 1)$

$y = - 2 \tan^{- 1} (\cos e c B) - \tan^{- 1} (\frac{2 \sin B}{\cos^{2} B})$

$= - (π + \tan^{- 1} \frac{2 \cos B}{1 - c o s e c^{2} B}) - \tan^{- 1} \frac{2 \sin B}{\cos^{2} B}$

$= - π$

Q 34 :

$If \sum_{r = 1}^{n} (\frac{\tan 2^{r - 1}}{\cos 2^{r}}) = \tan p^{n} - \tan q, then find the value of (p + q)$

(3)

$\sum_{r = 1}^{n} \frac{\sin (2^{r} - 2^{r - 1})}{\cos 2^{r} \cos 2^{r - 1}} = \sum_{r = 1}^{n} (\tan 2^{r} - \tan 2^{r - 1}) = \tan 2^{n} - \tan 1$

Q 35 :

Let $p = \tan (\frac{5 π}{9} \cos (2 \sin^{- 1} \frac{1}{\sqrt{5}})),$ $q = \sin^{- 1} (\sin \frac{2 π}{3}) + \cos^{- 1} (\cos (\frac{7 π}{6}))$ then the quadratic equation whose roots are $p$ , $\sec q$ is (Here inverse trigonometric functions take principal values)

$\sqrt{3} x^{2} - x + \sqrt{3} = 0$
$\sqrt{3} x^{2} - 4 x + 2 \sqrt{3} = 0$
$\sqrt{3} x^{2} - x - 2 \sqrt{3} = 0$
$x^{2} - \sqrt{3} x + = 0$

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$\cos (2 \sin^{- 1} \frac{1}{\sqrt{5}}) = \frac{3}{5},$ $\sin^{- 1} (\sin \frac{2 π}{3}) = \frac{π}{3},$ $\cos^{- 1} (\cos \frac{7 π}{6}) = 2 π - \frac{7 π}{6}$

Q 36 :

Exhaustive value of $x$ such that $\cos^{- 1} (\frac{8 x}{1 + 16 x^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

$(\frac{1}{8}, \infty)$
$(\frac{1}{16}, \frac{1}{8})$
$[\frac{1}{4}, \infty)$
$(\frac{1}{8}, \frac{1}{4})$

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(3)

$On solving \frac{π}{2} - \sin^{- 1} (\frac{2 \times 4 x}{1 + {(4 x)}^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

$\Rightarrow 2 \tan^{- 1} (4 x) = π - \sin^{- 1} (\frac{2 (4 x)}{1 + {(4 x)}^{2}})$

$4 x \geq 1; x \geq \frac{1}{4}$

Q 37 :

Which of the following is true?

$\frac{d y}{d x} for y = \sin^{- 1} (\cos x), where x \in (0, π), is - 1$
$\frac{d y}{d x} for y = \sin^{- 1} (\cos x), where x \in (0, 2 π), is 1$
$\frac{d y}{d x} for y = \cos^{- 1} (\sin x), where x \in (- π, \frac{π}{2}), is - 1$
$\frac{d y}{d x} for y = \cos^{- 1} (\sin x), where x \in (- \frac{π}{2}, \frac{3 π}{2}), is - 1$

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$Use graphs of \sin^{- 1} (\sin x) and \cos^{- 1} (\cos x)$

Topic Question Set

Q 21 :

Let $a \leq \tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x \leq b .$ If $α$ and $β$ denote the minimum and maximum possible values of $a$ and $b$ respectively, then:

Q 22 :

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

Q 23 :

The domain of the function defined by $f (x) = \sin^{- 1} x + \cos x$ is:

Q 24 :

The principal value of $\sec^{- 1} (\frac{- 2}{\sqrt{3}})$ is:

Q 25 :

If $\sin^{- 1} x = y,$ then:

Q 26 :

The domain of the function $\cos^{- 1} (2 x - 1)$ is:

Q 27 :

The value of $\cos^{- 1} (\cos \frac{3 π}{2})$ is equal to:

Q 28 :

The value of $\sin [2 \sin^{- 1} (0.6)]$ is:

Q 29 :

If $\cot^{- 1} (- \frac{1}{5}) = θ$ , the value of $\sin θ$ is:

Q 30 :

$\tan^{- 1} \sqrt{3} - \sec^{- 1} (- 2)$ is equal to:

Q 31 :

The value of $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$ is:

Q 33 :

If the equation $x^{3} + b x^{2} + c x + 1 = 0 (b < c)$ has only one real root $α$ . Then the value of $2 \tan^{- 1} (cosec α) + \tan^{- 1} (2 \sin α \sec^{2} α)$ is:

Q 34 :

$If \sum_{r = 1}^{n} (\frac{\tan 2^{r - 1}}{\cos 2^{r}}) = \tan p^{n} - \tan q, then find the value of (p + q)$

Q 35 :

Let $p = \tan (\frac{5 π}{9} \cos (2 \sin^{- 1} \frac{1}{\sqrt{5}})),$ $q = \sin^{- 1} (\sin \frac{2 π}{3}) + \cos^{- 1} (\cos (\frac{7 π}{6}))$ then the quadratic equation whose roots are $p$ , $\sec q$ is (Here inverse trigonometric functions take principal values)

Q 36 :

Exhaustive value of $x$ such that $\cos^{- 1} (\frac{8 x}{1 + 16 x^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

Q 37 :

Which of the following is true?

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	List I		List II
A.	The range of $\sin^{- 1} x$ is	I.	$[\frac{- π}{2}, \frac{π}{2}] - {0}$
B.	The range of $\tan^{- 1} x$ is	II.	$(\frac{π}{2}, \frac{π}{2})$
C.	The range of ${cosec}^{- 1} x$ is	III.	$[0, π] - {\frac{π}{2}}$
D.	The range of $\sec^{- 1} x$ is	IV.	$[\frac{- π}{2}, \frac{π}{2}]$

Topic Question Set

Q 21 : Let a≤tan-1x+cot-1x+sin-1x≤b. If α and β denote the minimum and maximum possible values of a and b respectively, then:

Q 22 : The principal value of sin-1(-32) is:

Q 23 : The domain of the function defined by f(x)=sin-1x+cosx is:

Q 24 : The principal value of sec-1(-23) is:

Q 25 : If sin-1x=y, then:

Q 26 : The domain of the function cos-1(2x-1) is:

Q 27 : The value of cos-1(cos3π2) is equal to:

Q 28 : The value of sin[2sin-1(0.6)] is:

Q 29 : If cot-1(-15)=θ, the value of sinθ is:

Q 30 : tan-13-sec-1(-2) is equal to:

Q 31 : The value of tan[12cos-1(53)] is:

Q 32 : Match List I with List II. List I List II A. The range of sin-1x is I. [-π2,π2]-{0} B. The range of tan-1x is II. (π2,π2) C. The range of cosec-1x is III. [0,π]-{π2} D. The range of sec-1x is IV. [-π2,π2] Choose the correct answer from the option given below.

Q 33 : If the equation x3+bx2+cx+1=0 (b<c) has only one real root α. Then the value of 2tan-1(cosec α)+tan-1(2sin α sec2 α) is:

Q 34 : If ∑r=1n(tan2r-1cos2r)=tanpn-tanq, then find the value of (p+q)

Q 35 : Let p=tan(5π9cos(2sin-115)), q=sin-1(sin2π3)+cos-1(cos(7π6)) then the quadratic equation whose roots are p, sec q is (Here inverse trigonometric functions take principal values)

Q 36 : Exhaustive value of x such that cos-1(8x1+16x2)=-π2+2tan-1(4x)

Q 37 : Which of the following is true?

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Q 21 :

Let $a \leq \tan^{- 1} x + \cot^{- 1} x + \sin^{- 1} x \leq b .$ If $α$ and $β$ denote the minimum and maximum possible values of $a$ and $b$ respectively, then:

Q 22 :

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

Q 23 :

The domain of the function defined by $f (x) = \sin^{- 1} x + \cos x$ is:

Q 24 :

The principal value of $\sec^{- 1} (\frac{- 2}{\sqrt{3}})$ is:

Q 25 :

If $\sin^{- 1} x = y,$ then:

Q 26 :

The domain of the function $\cos^{- 1} (2 x - 1)$ is:

Q 27 :

The value of $\cos^{- 1} (\cos \frac{3 π}{2})$ is equal to:

Q 28 :

The value of $\sin [2 \sin^{- 1} (0.6)]$ is:

Q 29 :

If $\cot^{- 1} (- \frac{1}{5}) = θ$ , the value of $\sin θ$ is:

Q 30 :

$\tan^{- 1} \sqrt{3} - \sec^{- 1} (- 2)$ is equal to:

Q 31 :

The value of $\tan [\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})]$ is:

Q 33 :

If the equation $x^{3} + b x^{2} + c x + 1 = 0 (b < c)$ has only one real root $α$ . Then the value of $2 \tan^{- 1} (cosec α) + \tan^{- 1} (2 \sin α \sec^{2} α)$ is:

Q 34 :

$If \sum_{r = 1}^{n} (\frac{\tan 2^{r - 1}}{\cos 2^{r}}) = \tan p^{n} - \tan q, then find the value of (p + q)$

Q 35 :

Let $p = \tan (\frac{5 π}{9} \cos (2 \sin^{- 1} \frac{1}{\sqrt{5}})),$ $q = \sin^{- 1} (\sin \frac{2 π}{3}) + \cos^{- 1} (\cos (\frac{7 π}{6}))$ then the quadratic equation whose roots are $p$ , $\sec q$ is (Here inverse trigonometric functions take principal values)

Q 36 :

Exhaustive value of $x$ such that $\cos^{- 1} (\frac{8 x}{1 + 16 x^{2}}) = - \frac{π}{2} + 2 \tan^{- 1} (4 x)$

Q 37 :

Which of the following is true?