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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Re-Attempt Weekly Quiz Competition

Introduction to Trigonometry Test - 56

Result

Introduction to Trigonometry Test - 56

Score

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Rank

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SHARING IS CARING

Question 1

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

$$1$$

$$0$$

$$-1$$

Solution

Question 2

$$\dfrac {1}{16}$$

$$0$$

$$1$$

None of these

Solution

Question 3

$$2\sec \theta$$

$$2 cosec \theta$$

$$0$$

$$1$$

Solution

Question 4

$$24^{\circ}$$

$$27^{\circ}$$

$$35^{\circ}$$

$$37^{\circ}$$

Solution

Question 5

$$1$$

$$-1$$

$$0$$

None of above

Solution

Question 6

$$1$$

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

$$\sqrt {3}$$

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

Solution

Question 7

$$\sqrt {3}$$

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

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

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

Solution

Question 8

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

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

$$\sqrt {3}$$

$$1$$

Solution

Question 9

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

$$\sin 15 + \cos 15$$

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

None of these

Solution

Question 10

$$0$$

$$1$$

$$2$$

$$-2$$

Solution

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