Class 10 maths Introduction to Trigonometry Hots Questions

Q 1 :

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

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

1

2

3

4

(1) $\frac{7}{25}$

Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

1/6
1/7
1/4
1/5

1

2

3

4

(1) 1/6

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

0°
30°
60°
90°

1

2

3

4

(2) 30°

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

1
2
3
4

1

2

3

4

(2)

$\sin θ + \cos θ = \sqrt{2}$

Squaring on both sides, we get

$\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ = 2$

$\Rightarrow 1 + 2 \sin θ \cos θ = 2$ $[∵ \sin^{2} θ + \cos^{2} θ = 1]$

$\Rightarrow 2 \sin θ \cos θ = 1 \Rightarrow \sin θ \cos θ = \frac{1}{2}$

But $\tan θ + \cot θ = \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} = \frac{\sin^{2} θ + \cos^{2} θ}{\cos θ \sin θ} = 2$

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

1/3
2/5
3/5
6

1

2

3

4

(1)

$5 \tan β = 4 \Rightarrow \tan β = \frac{4}{5}$

$∴$ $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} = \frac{5 \tan β - 2}{5 \tan β + 2}$ $[dividing \cos β by Nr. and Dr.]$

$= \frac{5 \times \frac{4}{5} - 2}{5 \times \frac{4}{5} + 2} = \frac{2}{6} = \frac{1}{3}$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

cos30°
tan30°
sin30°
cot30°

1

2

3

4

(2)

$x \tan 60 ° \cos 60 ° = \sin 60 ° \cot 60 °$

$\Rightarrow x \times \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{3}} \Rightarrow x \sqrt{3} = 1$

$\Rightarrow x = \frac{1}{\sqrt{3}} \Rightarrow x = \tan 30 °$ $[∵ \tan 30 ° = \frac{1}{\sqrt{3}}]$

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

$1 - \frac{1}{m}$
$m^{2} - 1$
$\frac{1}{m}$
$- m$

1

2

3

4

(3)

Given, $\sec θ - \tan θ = m$ ...(i)

We know that, $\sec^{2} θ - \tan^{2} θ = 1$

$\Rightarrow (\sec θ - \tan θ) (\sec θ + \tan θ) = 1$

$\Rightarrow m (\sec θ + \tan θ) = 1$

$\Rightarrow \sec θ + \tan θ = \frac{1}{m}$

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

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

1

2

3

4

(1)

$\cos (α + β) = 0 = \cos 90 ° \Rightarrow (α + β) = 90 ° \Rightarrow \frac{α + β}{2} = 45 °$

$\Rightarrow \cos (\frac{α + β}{2}) = \cos 45 ° = \frac{1}{\sqrt{2}}$

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is:

0
2
1
– 1

1

2

3

4

(3) 1

Q 10 :

If $\tan^{2} θ + \cot^{2} α = 2,$ where $θ = 45 ° and θ ° \leq α \leq 90 °,$ then the value of a is :

30°
45°
60°
90°

1

2

3

4

(2) 45°

Topic Question Set

Q 1 :

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

Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is:

Q 10 :

If $\tan^{2} θ + \cot^{2} α = 2,$ where $θ = 45 ° and θ ° \leq α \leq 90 °,$ then the value of a is :

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Topic Question Set

Q 2 : If 5 tan θ = 4, then the value of 5sinθ-3cosθ5sinθ+2cosθ 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 3 : Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) 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 4 : If sinθ + cosθ = 2, then tanθ + cot θ = .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 5 : If 5 tanβ = 4, then 5sinβ-2cosβ5sinβ+2cosβ= .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 6 : If x tan 60°cos 60°= sin60°cot 60°, 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 7 : If sec θ – tan θ = m, then the value of sec θ + tan 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 8 : If cos (α+β) = 0, then value of cos (α+β2) 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 9 : The value of (sin2θ+11+tan2θ) 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; }

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Q 2 :

If 5 tan θ = 4, then the value of $\frac{5 \sin θ - 3 \cos θ}{5 \sin θ + 2 \cos θ}$ is

Q 3 :

Given that sin α = 1/2 and cos β = 1/2, then the value of (β – α) is

Q 4 :

If sinθ + cosθ = $\sqrt{2}$ , then tanθ + cot θ =

Q 5 :

If 5 tanβ = 4, then $\frac{5 \sin β - 2 \cos β}{5 \sin β + 2 \cos β} =$

Q 6 :

If x tan 60°cos 60°= sin60°cot 60°, then x =

Q 7 :

If sec θ – tan θ = m, then the value of sec θ + tan θ is :

Q 8 :

If cos $(α + β)$ = 0, then value of cos $(\frac{α + β}{2})$ is equal to:

Q 9 :

The value of $(\sin^{2} θ + \frac{1}{1 + \tan^{2} θ})$ is: