Question

Topic - Trigonometry✔️

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Answer 1

Answer:

1. In a right angled triangle ABC :-

∠B = 90°

tan A = 5/ 12

We know that, tan Ф = perpendicular(p)/ base(b)

tan A = p/ b

perpendicular in the given triangle is the side AB

and base is the side BC

tan A = AB/ BC

$\frac{5}{12} = \frac{AB}{BC} = \frac{P}{B}$

AB = perpendicular = 5

BC = base = 12

Using Pythagoras theorem :-

H² = P² + B²

H² = 5² + 12²

H² = 25 + 144

H² = 169

H = √169

H = 13 (h = hypotenuse)

AC = hypotenuse = 13

sin A = perpendicular/ hypotenuse

sin A= 5/ 13

Hence, option (2) sinA = 5/ 13 is correct

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2. In a triangle :-

tan ∅ = 5/ 12

(from the above question we know that p = 5, b = 12, h = 13)

perpendicular = 5

base = 12

hypotenuse = 13

sec ∅ = hypotenuse/ base

= 13/ 5

cosec ∅ = hypotenuse/ perpendicular

= 13/ 12

sec∅ + cosec∅ = 13/ 5 + 13/ 12

= (13 x 5 + 13 x 12)/ 12 x 5

= (65 + 156)/ 60

= 221/ 60

Hence, option (2) 221/ 60 is correct

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3. 4cot A = 3

cot A = 3/ 4

cot A = base/ perpendicular

hence, base = 3

perpendicular = 4

Using Pythagoras theorem :-

H² = P² + B²

H² = 4² + 3²

H² = 16 + 9

H² = 25

H = √25

H = 5 (h = hypotenuse)

sin A = perpendicular/ hypotenuse

= 4/ 5

cos A = base/ hypotenuse

= 3/ 5

= $\frac{sin A + cos A }{sin A - cos A }$

$= \frac{4/ 5 + 3/ 5}{4/ 5 - 3/ 5}$

= $\frac{7/ 5}{1/ 5}$

= 7

Hence, option(1) 7 is correct

method II

$\frac{sin A + cos A }{sin A - cos A }$

dividing the eqn. with sin A :-

= $\frac{(sin A/ sin A) + (cos A/ sin A)}{(sin A/ sin A) - (cos A/ sin A)}$

= $\frac{1 + cot A}{1 - cot A}$ (as cos A/ sin A = cot A)

= $\frac{1 + 3/ 4}{1 - 3/4}$

= $\frac{(4 + 3)/ 4}{(4 - 3)/ 4}$

= 7

Answer 2

Question 1 :

In a right angled triangle ABC, ∠B = 90° and tanA = 5/12, then?

1) cotA = 5/13  2) sinA = 5/13  3) sinA = -5/13  4) cosA = 13/5

Solution :

tanA = 5/12

∵ tanA = perpendicular/base

∴ Perpendicular of ΔABC = 5

∴ Base of ΔABC = 12

∴ Hypotenuse = √(12² + 5²)

⇒ Hypotenuse = √144 + 25

⇒ Hypotenuse = √169

⇒ Hypotenuse = 13

∴ cotA = 1/tanA = 1/(5/12) = 12/5

∴ 1st option is incorrect.

∵ sinA = perpendicular/hypotenuse

⇒ sinA = 5/13

∴ 2nd option (sinA = 5/13) is correct

[Note : Other options are also incorrect.]

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Question 2 :

If tanФ = 5/12 and Ф is an acute angle, then secФ + cosecФ = ?

1) 22/60   2) 221/60   3)22/7  4)None

Solution :

tanФ = 5/12

Perpendicular = 5

Base = 12

From pythagoras triplet, hypotenuse = 13

∵ secФ = hypotenuse/base

Also, cosecФ = hypotenuse/perpendicular

∴ secФ + cosecФ = 13/12 + 13/5

⇒ secФ + cosecФ = (65 + 156)/60

⇒ secФ + cosecФ = 221/60

∴ 2nd option (221/60) is correct.

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Question 3 :

4cotA = 3, then value of (sinA + cosA)/(sinA - cosA) = ?

1) 7    2) 2/11    3) 1/2   4) 1

Solution :

4cotA = 3

∴ cotA = 3/4

∵ cotA = base/perpendicular

∴ Base = 3

And, perpendicular = 4

Now acc. to pythagorian triplet, hypotenuse = 5

∴ sinA = 4/5

∴ cosA = 3/5

Now given expression :

⇒ (sinA + cosA)/(sinA - cosA)

⇒ (4/5 + 3/5)/(4/5 - 3/5)

⇒ {(4 + 3)/5}/{(4 - 3)/5}

⇒ (7/5)/(1/5)

⇒ (7/5) × 5

⇒ 7

∴ Required value of the expression = 7

∴ 1st option (7) is correct.