Class 10 maths Introduction to Trigonometry Hots Questions

Q 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

$1 / \sqrt{2}$
$\sqrt{2}$
$1$
$0$

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(3)

$\sin θ = \cos θ \Rightarrow \tan θ = 1 = 45 ° \Rightarrow θ = 45 °$

Now, $\sec θ . \sin θ = \sec 45 ° . \sin 45 ° = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1$

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

$\frac{5}{12}$
$\frac{12}{13}$
$\frac{5}{13}$
$\frac{12}{5}$

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(2)

$G i v e n, 5 t a n θ - 12 = 0 \Rightarrow t a n θ = \frac{12}{5} = \frac{p}{b} L e t p = 12 k, b = 5 k h = \sqrt{p^{2} + b^{2}} = \sqrt{(12 k)^{2} + (5 k)^{2}} = \sqrt{144 k^{2} + 25 k^{2}} = \sqrt{169 k^{2}} = 13 k si n θ = \frac{p}{h} = \frac{12 k}{13 k} = \frac{12}{13}$

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

$\frac{7}{25}$
$\frac{24}{25}$
$\frac{7}{24}$
$\frac{24}{7}$

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(1)

$s i n A = \frac{p}{h} = \frac{B C}{A C} = \frac{7}{25} c o s C = \frac{b}{h} = \frac{B C}{A C} = \frac{7}{25}$

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

2
1/2
1/3
1/4

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(2)

$F r o m t h e g i v e n f i g u r e \frac{\cot y}{\cot x} = \frac{A C}{B C} \div \frac{A C}{C D} = \frac{A C}{B C} \times \frac{C D}{A C} = \frac{C D}{B C} = \frac{C D}{2 C D} = \frac{1}{2} (U s i n g c o t θ = \frac{b}{p})$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

$\frac{b - a}{b^{2} + a^{2}}$
$\frac{b + a}{b^{2} + a^{2}}$
$\frac{b^{2} + a^{2}}{b^{2} - a^{2}}$
$\frac{b^{2} - a^{2}}{b^{2} + a^{2}}$

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(4)

$W e h a v e, t a n θ = \frac{b}{a} G i v e n e x p r e s s i o n i s \frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ} T a k i n g \cos θ c o m m o n f r o m t h e e x p r e s s i o n, i t g i v e s \frac{b \tan θ - a}{b \tan θ + a} \dots (i) S u b s t i t u t i n g t a n θ = \frac{b}{a} i n e q u a t i o n (i) g i v e s \frac{\frac{b^{2}}{a} - a}{\frac{b^{2}}{a} + a} = \frac{b^{2} - a^{2}}{b^{2} + a^{2}}$

Q 16 :

Which of the following trigonometric ratios are correctly written in context of $∆ A B C a s g i v e n ?$

$(i) s i n A = \frac{B C}{A C} (i i) t a n C = \frac{B C}{A B} (i i i) c o s C = \frac{A B}{A C} (i v) s e c A = \frac{A C}{A B}$

(i) and (ii)
(ii) and (iv)
(i) and (iv)
(ii) and (iii)

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(3)

$W h e n ∠ A i s u n d e r c o n s i d e r a t i o n B a s e = A B, P e r p e n d i c u l a r = B C, H y p o t e n u s e = A C ∴ s i n A = \frac{Perpendicular}{Hypotenuse} = \frac{B C}{A C} and s e c A = \frac{Hypotenuse}{Base} = \frac{A C}{A B} W h e n ∠ C i s u n d e r c o n s i d e r a t i o n : B a s e = B C, P e r p e n d i c u l a r = A B, H y p o t e n u s e = A C ∴ t a n C = \frac{Perpendicular}{Base} = \frac{A B}{B C} and c o s C = \frac{Base}{Hypotenuse} = \frac{B C}{A C}$

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

$\sqrt{3}$
$\frac{1}{\sqrt{3}}$
1
0

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(3)

Given

$s i n α = \frac{\sqrt{3}}{2} = s i n 60 ° \Rightarrow α = 60 ° A n d, c o s β = \frac{\sqrt{3}}{2} = c o s 30 ° \Rightarrow β = 30 ° ∴ t a n α \cdot t a n β = t a n 60 ° \cdot t a n 30 ° = \sqrt{3} \times \frac{1}{\sqrt{3}} = 1$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

$30 °$
$60 °$
$45 °$
$90 °$

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(2)

$2 {s i n}^{2} θ = \frac{1}{2} \Rightarrow {s i n}^{2} θ = \frac{1}{4} \Rightarrow s i n θ = \frac{1}{2} = s i n 30 ° \Rightarrow θ = 30 °$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

$- \frac{5}{3}$
$- \frac{1}{2}$
0
$\frac{1}{4}$

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(3)

We have,

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °] = \frac{5}{8} \times (2)^{2} - (\sqrt{3})^{2} + {(\frac{1}{\sqrt{2}})}^{2} = \frac{5}{8} \times 4 - 3 + \frac{1}{2} = \frac{5}{2} - 3 + \frac{1}{2} = \frac{5 - 6 + 1}{2} = 0$

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to

$\sin 60 °$
$\cos 60 °$
$\tan 60 °$
$\sin 30 °$

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(1)

We have,

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °} = \frac{2 \times \frac{1}{\sqrt{3}}}{1 + {(\frac{1}{\sqrt{3}})}^{2}} = \frac{\frac{2}{\sqrt{3}}}{1 + \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{4}{3}} = \frac{2}{\sqrt{3}} \times \frac{3}{4} = \frac{\sqrt{3}}{2} = s i n 60 °$

Topic Question Set

Q 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

Q 16 :

Which of the following trigonometric ratios are correctly written in context of $∆ A B C a s g i v e n ?$

$(i) s i n A = \frac{B C}{A C} (i i) t a n C = \frac{B C}{A B} (i i i) c o s C = \frac{A B}{A C} (i v) s e c A = \frac{A C}{A B}$

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to

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Topic Question Set

Q 11 : If sinθ=cosθ, (0°<θ<90°), then value of (secθ.sinθ) 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 12 : If 5 tan θ-12=0, then the value of sin θ 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 13 : In ∆ABC, right angled at B sinA=725, then the value of cos C 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 14 : In the given figure, D is the mid-point of BC , then the value of cotycotx 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 15 : If a tanθ=b the value of bsinθ-acosθbsinθ+acosθ 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 17 : If sinα=32,cosβ=32, then tanα·tanβ 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 18 : The value of θ for which 2sin 2θ=12,0°≤θ≤90° 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 19 : 58sec 260°-tan 260°+cos 245° 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; }

Q 20 : 2tan30°1+tan 230° 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 11 :

If $\sin θ = \cos θ,$ $(0 ° < θ < 90 °),$ then value of $(\sec θ . \sin θ)$ is:

Q 12 :

If $5 t a n θ - 12 = 0$ , then the value of $\sin θ$ is

Q 13 :

In $∆ A B C$ , right angled at B $s i n A = \frac{7}{25}$ , then the value of cos C is

Q 14 :

In the given figure, D is the mid-point of BC , then the value of $\frac{\cot y}{\cot x} i s$

Q 15 :

If $a t a n θ = b$ the value of $\frac{b \sin θ - a \cos θ}{b \sin θ + a \cos θ}$ is

Q 17 :

If $s i n α = \frac{\sqrt{3}}{2}, c o s β = \frac{\sqrt{3}}{2}, t h e n t a n α \cdot t a n β i s :$

Q 18 :

The value of $θ$ for which $2 {s i n}^{2} θ = \frac{1}{2}, 0 ° \leq θ \leq 90 ° i s$

Q 19 :

$[\frac{5}{8} {s e c}^{2} 60 ° - {t a n}^{2} 60 ° + {c o s}^{2} 45 °]$ is equal to

Q 20 :

$\frac{2 t a n 30 °}{1 + {t a n}^{2} 30 °}$ is equal to