Class 10 maths Introduction to Trigonometry Hots Questions

Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

1
40/9
38/9
$5 \frac{1}{3}$

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4

(4)

Given,

$c o t θ = \frac{1}{\sqrt{3}} = c o t 60 ° \Rightarrow θ = 60 ° N o w, {s e c}^{2} θ + {c o s e c}^{2} θ = {s e c}^{2} 60 ° + {c o s e c}^{2} 60 ° = (2)^{2} + {(\frac{2}{\sqrt{3}})}^{2} = 4 + \frac{4}{3} = \frac{12 + 4}{3} = \frac{16}{3} = 5 \frac{1}{3}$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

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

1

2

3

4

(1)

Given,

$s e c θ = \sqrt{2} \Rightarrow θ = 45 ° N o w, \frac{1 + t a n θ}{\sin θ} = \frac{1 + t a n 45 °}{\sin 45 °} = \frac{1 + 1}{\frac{1}{\sqrt{2}}} = \frac{2}{\frac{1}{\sqrt{2}}} = 2 \sqrt{2}$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

$\cos 30 °$
$\tan 30 °$
$\sin 30 °$
$\cot 30 °$

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(2)

Given expression is

$x \tan 60 ° c o s 60 ° = s i n 60 ° c o t 60 ° \Rightarrow x \times \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}} \Rightarrow x \times \frac{\sqrt{3}}{2} = \frac{1}{2} \Rightarrow x = \frac{1}{\sqrt{3}} = t a n 30 °$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

0
1
$\frac{2}{\sqrt{3}}$
not defined

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(4)

$I n Δ A B C, ∠ C = 90 ° (given) A n d, ∠ A + ∠ B + ∠ C = 180 ° \Rightarrow ∠ A + ∠ B = 90 ° o r A + B = 90 ° \Rightarrow s e c (A + B) = s e c 90 ° = not defined$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$

0
2
20
$2^{20}$

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(2)

We have,

$t a n α + c o t α = 2 \Rightarrow t a n α + \frac{1}{\tan α} = 2 \Rightarrow {t a n}^{2} α - 2 t a n α + 1 = 0 \Rightarrow (t a n α - 1)^{2} = 0 \Rightarrow t a n α = 1 \Rightarrow α = 45 ° S o, t a n α = c o t α = 1 N o w, {t a n}^{20} α + {c o t}^{20} α = 1^{20} + 1^{20} = 1 + 1 = 2$

Q 26 :

The value of which of the following is definitely zero?

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

both (i) and (iv)
both (i) and (iii)
only (i)
only (iii)

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(2)

$(i) s i n 45 ° = \frac{1}{\sqrt{2}}, c o s 45 ° = \frac{1}{\sqrt{2}} \Rightarrow s i n 45 ° - c o s 45 ° = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0 (i i) t a n 90 ° = \infty, c o t 90 ° = 0 \Rightarrow t a n 90 ° - c o t 90 ° = not defined (iii) \sin 60 ° = \frac{\sqrt{3}}{2}, \cos 30 ° = \frac{\sqrt{3}}{2} \Rightarrow \sin 60 ° - \cos 30 ° = \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 (iv) \tan 30 ° = \frac{1}{\sqrt{3}}, \cot 30 ° = \sqrt{3} \tan 30 ° - \cot 30 ° = \frac{1}{\sqrt{3}} - \sqrt{3} = \frac{1 - 3}{\sqrt{3}} = - \frac{2}{\sqrt{3}} \neq 0$

Hence, (i) and (iii) will definitely give zero.

Q 27 :

P and Q are acute angles such that P > Q. Then which of the following is definitely true?

$(i) s i n P < s i n Q (i i) t a n P > t a n Q (i i i) c o s P > c o s Q (i v) c o s P > s i n Q$

only (i)
only (ii)
(i) and (ii) both
only (iv)

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(2)

Given that P and Q are acute angles such that P > Q

$L e t P = 60^{°} Q = 30 ° . C h e c k i n g (i) : W e k n o w t h a t \sin θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° s i n 60 ° = \frac{\sqrt{3}}{2}, s i n 30 ° = \frac{1}{2} \Rightarrow s i n Q < s i n P T h u s (i) i s n o t t r u e C h e c k i n g (i i) : W e k n o w t h a t \tan θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° \Rightarrow t a n P > t a n Q H e n c e (i i) i s t r u e . C h e c k i n g (i i i) : W e k n o w t h a t \cos θ i n c r e a s e s a s θ i n c r e a s e s f r o m 0 ° t o 90 ° S o (i i i) i s n o t t r u e . C h e c k i n g (i v) : c o s P = c o s 60 ° = \frac{1}{2}, s i n Q = s i n 30 ° = \frac{1}{2} \Rightarrow c o s P = s i n Q$

Topic Question Set

Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$

Q 26 :

The value of which of the following is definitely zero?

$(i) s i n 45 ° - c o s 45 ° (i i) t a n 90 ° - c o t 90 ° (i i i) s i n 60 ° - c o s 30 ° (i v) t a n 30 ° - c o t 30 °$

Q 27 :

P and Q are acute angles such that P > Q. Then which of the following is definitely true?

$(i) s i n P < s i n Q (i i) t a n P > t a n Q (i i i) c o s P > c o s Q (i v) c o s P > s i n Q$

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Topic Question Set

Q 21 : If cotθ=13,the value of sec 2θ+cosec 2θ is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 22 : Given that secθ=2, the value of1+tanθsinθ is .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 23 : If x tan60°cos60°=sin60°cot60°,then x= .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 24 : If ΔABC is right angled at C, then the value of secA+Bis .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

Q 25 : If If tanα+cotα=2,then tan 20α+cot 20 α is equal to .question-row { display: flex; align-items: flex-start; gap: 8px; } .q-no { font-size: 17px; white-space: nowrap; margin-left: -15px; } .q-text { font-size: 18px; flex: 1; } .q-no, .q-text p { line-height: 30px; }

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Q 21 :

$I f c o t θ = \frac{1}{\sqrt{3}}, t h e v a l u e o f {s e c}^{2} θ + {c o s e c}^{2} θ i s$

Q 22 :

Given that $s e c θ = \sqrt{2}, t h e v a l u e o f \frac{1 + t a n θ}{\sin θ} i s$

Q 23 :

$I f x t a n 60 ° c o s 60 ° = s i n 60 ° c o t 60 °, t h e n x =$

Q 24 :

$I f Δ A B C$ is right angled at C, then the value of $s e c (A + B) i s$

Q 25 :

If $I f t a n α + c o t α = 2, t h e n {t a n}^{20} α + {c o t}^{20} α i s e q u a l t o$