Introduction to Trigonometry Test

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Introduction to Trigonometry Test - 56

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Question 1
1 / -0

$\cot 1^{\circ} \cot 2^{\circ} ...... \cot 89^{\circ} =$

A
$\dfrac {1}{2}$

B
$1$

C
$0$

D
$-1$

Solution

$\cot 1^{\circ} \cot 2^{\circ} ...... \cot 89^{\circ}$
$=(\cot 1^o\cdot \cot 89^o)(\cot 2^o\cdot \cot 88^o)...............(\cot 44^o\cdot \cot 46^o)(\cot 45^o)$
$=(\cot 1^o\cdot \tan 1^o)(\cot 2^o\cdot \tan 2^o)...............(\cot 44^o\cdot \tan 44^o)(\cot 45^o)$ , since $\cot (90^o-\theta)=\tan\theta$
$=(1)(1).......(1)(1)=1$ , since $\tan\theta\cdot \cot\theta=1$
Question 2
1 / -0

Find the value of $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$

A
$\dfrac {1}{16}$

B
$0$

C
$1$

D
None of these

Solution
Question 3
1 / -0

$\text{cosec } (75^{\circ} + \theta) - \sec (15^{\circ} - \theta) =$

A
$2\sec \theta$

B
$2 cosec \theta$

C
$0$

D
$1$

Solution

Given,
$\csc { \left( { 75^o } +\theta \right) } -\sec { \left( { 15^o } -\theta \right) }$
$=\csc { \left( { 75^o } +\theta \right) } -\sec { \left[ { { 90^o } -\left( 75^o+\theta \right) } \right] }$
$=\csc { \left( { 75^o } +\theta \right) } -\csc { \left( { 75^o } +\theta \right) }$
$=0$
$0$ will be the answer
Question 4
1 / -0

If $\tan 2A = \cot (A - 21^{\circ})$ , where $2A$ is an acute angle, then $\angle A =$ ____

A
$24^{\circ}$

B
$27^{\circ}$

C
$35^{\circ}$

D
$37^{\circ}$

Solution

Given, $\tan 2A = \cot (A - 21^{\circ})$
Therefore, $\cot (90^{\circ} - 2A) = \cot (A - 21^{\circ})$
$\Rightarrow 90^{\circ} - 2A = A - 21^{\circ}$
$\Rightarrow 90^{\circ} + 21^{\circ} = 2A + A$
$\Rightarrow 111^{\circ} = 3A$
$\Rightarrow A = 37^{\circ}$
So, $\angle A=37^{\circ}$ is correct.
Question 5
1 / -0

$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} ... \tan 89^{\circ} =$

A
$1$

B
$-1$

C
$0$

D
None of above

Solution

$(\tan 1^{\circ} \tan 89^{\circ}) (\tan 2^{\circ} \tan 88^{\circ}) ..... (\tan 44^{\circ} \tan 46^{\circ}) \tan 45^{\circ}$ , group pair of terms
$= \left \{\tan 1^{\circ} \tan (90^{\circ} - 1^{\circ})\right \} \left \{\tan 2^{\circ} \tan (90^{\circ} - 2^{\circ})\right \} ...... \left \{\tan 44^{\circ} \tan (90^{\circ} - 44^{\circ})\right \}\tan 45^{\circ}$
$= (\tan 1^{\circ} \cot 1^{\circ})(\tan 2^{\circ} \cot 2^{\circ}) ..... (\tan 44^{\circ} \cot 44^{\circ}) \tan 45^{\circ}$
$= (1 \times 1 ..... 1)\times 1 = 1$

Note that $\tan\theta\times \cot\theta=1$
Question 6
1 / -0

$\tan 5^{\circ} \tan 25^{\circ} \tan 30^{\circ} \tan 65^{\circ} \tan 85^{\circ} =$

A
$1$

B
$\dfrac {1}{2}$

C
$\sqrt {3}$

D
$\dfrac {1}{\sqrt {3}}$

Solution

$(\tan 5^{\circ} \tan 85^{\circ}) (\tan 25^{\circ} \tan 65^{\circ}) \tan 30^{\circ}$
$= [\tan 5^{\circ} \tan (90^{\circ} - 5^{\circ})] [\tan 25^{\circ} (\tan 90^{\circ} - 25^{\circ})] \times \tan 30^{\circ}$
$= \tan 5^{\circ} \cot 5^{\circ} \tan 25^{\circ} \cot 25^{\circ} \times \dfrac {1}{\sqrt {3}}$
$= 1\times 1\times \dfrac {1}{\sqrt {3}} = \dfrac {1}{\sqrt {3}}$ .
Question 7
1 / -0

Find the value of : $\dfrac {\cos 38^{\circ} \csc 52^{\circ}}{\tan 18^{\circ} \tan 35^{\circ} \tan 60^{\circ} \tan 72^{\circ} \tan 55^{\circ}} =$

A
$\sqrt {3}$

B
$\dfrac {1}{3}$

C
$\dfrac {1}{\sqrt {3}}$

D
$\dfrac {2}{\sqrt {3}}$

Solution

Given:
$\dfrac { \cos { { 38^o } \csc { { 52^o } } } }{ \tan { { 18^o } } \tan { { 35^o } \tan { { 60^o } } \tan { { 72^o } } \tan { { 55^o } } } }$

$=\dfrac { \cos { { 38^o } } \csc { \left( { 90^o- } { 38^o } \right) } }{ \left( \tan { { 18^o\times } } \tan { { 72^o } } \right) \times \left( \tan { { 35^o\times } } \tan { { 55^o } } \right) \times \tan { { 60^o } } }$

$=\dfrac { \cos { { 38^o } } \sec { { 38^o } } }{ \tan { { 18^o } } \tan { \left( { 90^o-18^o } \right) \tan { { 35^o } } \tan { \left( { 90^o- } { 35^o } \right) \tan { { 60^o } } } } }$

( we know that $\tan { \left( 90^o-\theta \right) } =\cot { \theta }$ and $\cot { \theta } =\dfrac { 1 }{ \tan { \theta } }$ and $\sec { \theta =\dfrac { 1 }{ \cos { \theta } } }$ and $\tan { { 60^o=\sqrt { 3 } } } )$

$\therefore \dfrac { \cos { { 38^o } \csc { { 52^o } } } }{ \tan { { 18^o } } \tan { { 35^o } \tan { { 60^o } } \tan { { 72^o } } \tan { { 55^o } } } }$

$=\dfrac { 1 }{ \left( \tan { { 18^o } } \cot { { 18^o } } \right) \left( \tan { { 35^o } } \cot { { 35^o } } \right) \sqrt { 3 } }$

$=\dfrac { 1 }{ \sqrt { 3 } }$
Question 8
1 / -0

Evaluate: $\dfrac {\tan 30^{\circ}}{\cot 60^{\circ}}$

A
$\dfrac {1}{\sqrt {2}}$

B
$\dfrac {1}{\sqrt {3}}$

C
$\sqrt {3}$

D
$1$

Solution

Taking $\dfrac {\tan 30^{\circ}}{\cot 60^{\circ}}$
$\dfrac {\tan 30^{\circ}}{\cot (90-30)^{\circ}}$
as we know, $\cot { \left( 90^0-\theta \right) } =\tan { \theta }$
$\dfrac {\tan 30^{\circ}}{\tan30^{\circ}}$
$=1$
Question 9
1 / -0

$\cos 75^{\circ} + \cot 75^{\circ}$ , when expressed in terms of angles between $0^{\circ}$ and $30^{\circ}$ , becomes

A
$\sin 15^{\circ} + \tan 15^{\circ}$

B
$\sin 15 + \cos 15$

C
$\cos 15^{\circ} + \tan 15^{\circ}$

D
None of these

Solution

We know that
$\cos A = \sin(90˚-A)$
$\cot A = \tan(90˚-A)$
$\text{cosec } A = \sec(90˚-A)$

Using these, we get
$\cos 75^{\circ} = \sin (90˚ - 75˚) = \sin 15^{\circ}$
$\cot 75^{\circ} = \tan (90˚ - 75˚) = \tan 15^{\circ}$

$\therefore \cos 75^{\circ} + \cot 75^{\circ} = \sin 15^{\circ} + \tan 15^{\circ}$ .
Question 10
1 / -0

Without trigonometric table, evaluate $\dfrac {\cot 63^{\circ}}{\tan 27^{\circ}}$

A
$0$

B
$1$

C
$2$

D
$-2$

Solution

Taking
$\dfrac {\cot 63^{\circ}}{\tan 27^{\circ}}$
$\dfrac {\cot (90-27)^{\circ}}{\tan 27^{\circ}}$
We know,
$\cot { \left( 90^0-\theta \right) } =\tan { \theta }$
$\dfrac {\tan27^{\circ}}{\tan 27^{\circ}}$
$=1$

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Introduction to Trigonometry Test - 56

Result

Introduction to Trigonometry Test - 56

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SHARING IS CARING

Question 1

12\dfrac {1}{2}21​

111

000

−1-1−1

Solution

Question 2

116\dfrac {1}{16}161​

000

111

None of these

Solution

Question 3

2sec⁡θ2\sec \theta2secθ

2cosecθ2 cosec \theta2cosecθ

000

111

Solution

Question 4

24∘24^{\circ}24∘

27∘27^{\circ}27∘

35∘35^{\circ}35∘

37∘37^{\circ}37∘

Solution

Question 5

111

−1-1−1

000

None of above

Solution

Question 6

111

12\dfrac {1}{2}21​

3\sqrt {3}3​

13\dfrac {1}{\sqrt {3}}3​1​

Solution

Question 7

3\sqrt {3}3​

13\dfrac {1}{3}31​

13\dfrac {1}{\sqrt {3}}3​1​

23\dfrac {2}{\sqrt {3}}3​2​

Solution

Question 8

12\dfrac {1}{\sqrt {2}}2​1​

13\dfrac {1}{\sqrt {3}}3​1​

3\sqrt {3}3​

111

Solution

Question 9

sin⁡15∘+tan⁡15∘\sin 15^{\circ} + \tan 15^{\circ}sin15∘+tan15∘

sin⁡15+cos⁡15\sin 15 + \cos 15sin15+cos15

cos⁡15∘+tan⁡15∘\cos 15^{\circ} + \tan 15^{\circ}cos15∘+tan15∘

None of these

Solution

Question 10

000

111

222

−2-2−2

Solution

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$\dfrac {1}{2}$

$1$

$0$

$-1$

$\dfrac {1}{16}$

$0$

$1$

$2\sec \theta$

$2 cosec \theta$

$0$

$1$

$24^{\circ}$

$27^{\circ}$

$35^{\circ}$

$37^{\circ}$

$1$

$-1$

$0$

$1$

$\dfrac {1}{2}$

$\sqrt {3}$

$\dfrac {1}{\sqrt {3}}$

$\sqrt {3}$

$\dfrac {1}{3}$

$\dfrac {1}{\sqrt {3}}$

$\dfrac {2}{\sqrt {3}}$

$\dfrac {1}{\sqrt {2}}$

$\dfrac {1}{\sqrt {3}}$

$\sqrt {3}$

$1$

$\sin 15^{\circ} + \tan 15^{\circ}$

$\sin 15 + \cos 15$

$\cos 15^{\circ} + \tan 15^{\circ}$

$0$

$1$

$2$

$-2$