Trigonometric Ratios & Identity Model Questions Set 1 Practice Questions Answers Test With Solutions & More Shortcuts

TRIGONOMETRIC RATIOS & IDENTITIES PRACTICE TEST [1 - EXERCISES]

Trigonometric Ratios & Identity Model Questions Set 1

Question : 1

Consider the following :

  1. ${cos 75°}/{sin 15 °} + {sin 12°}/{cos 78°} - {cos 18°}/{sin 72°}$ = 1
  2. ${cos 35°}/{sin 55°} - {sin 11°}/{cos 79°}$ + cos 28° cosec 62°= l
  3. ${sin 80°}/{cos 10°}$ - sin 59° sec 31° = 0
Which of the above are correct?

a) 2 and 3 only

b) 1 and 2 only

c) 1 and 3 only

d) 1, 2 and 3

Answer: (d)

1. ${cos 75°}/{sin 15°} + {sin 12°}/{cos 78°} - {cos 18°}/{sin 72°}$ = 1

${cos 75°}/{cos 75°} + {sin 12°}/{sin 12°} - {cos 18°}/{cos 18°}$ = 1

2 - 1 = 1

Statement is correct.

2. ${cos 35°}/{sin 55°} - {sin 11°}/{cos 79°}$ + cos 28° cosec 62° = 1

${cos 35°}/{cos 35°} - {sin 11°}/{sin 11°}$ + cos 28° sec 28° = 1

1 - 1 + 1 = 1

2nd Statement is also true.

3. ${sin 80°}/{cos 10°}$ - sin 59° sec 31° = 0

${sin 80°}/{sin 80°}$ - sin 59° cosec 59° = 0

1 - 1 = 0

3rd Statement is correct.

Question : 2

If cos A = tan B, cos B = tan C and cos C = tan A then sin A is equal to

a) ${√5 - 1}/2$

b) ${√5 - 1}/4$

c) ${√3 - 1}/4$

d) None of these

Answer: (d)

Cos A = tan B

Squaring on both sides

$cos^2 A = tan^2 B$

⇒ $tan^2 B = {sin^2 B}/{cos^2 B} = {1 - cos^2 B}/{cos^2 B}$

∴ $cos^2 A = {1 - cos^2 B}/{cos^2 B}$

$cos^2 A = {1 - tan^2 C}/{tan^2 C}$

(∵ cos B = tan C)

⇒ $cos^2 A tan^2 C = 1 - tan^2 C = 1 - {sin^2 C}/{cos^2 C} $

⇒ $cos^2 A {sin^2 C}/{cos^2 C} ={cos^2 C - sin^2 C}/{cos^2 C}$

⇒ $cos^2 A (1 - cos^2 C) = 2 cos^2 C - 1$

⇒ $cos^2 A (1 - tan^2 A) = 2 tan^2 A - 1$

$cos^2 A - sin^2 A = {2 sin^2 A}/{cos^2 A} - 1$

⇒ 1 - $2sin^2 A = {2 sin^2 A - cos^2 A}/{cos^2 A}$

⇒ $cos^2 A (1 - sin^2 A) = 2 sin^2 A - cos^2 A$

⇒ $cos^2 A (1 - 2 sin^2 A) = 2 sin^2 A - 1 + sin^2 A$

⇒ $(1 - sin^2 A) (1 - 2 sin^2 A) = 3 sin^2 A - 1$

⇒ $1 - 2 sin^2 A - sin^2 A + 2 sin^4A = 3 sin^2 A - 1$

⇒ $1 - 3 sin^2 A + 2 sin^4 A = 3 sin^2 A - 1$

⇒ $2 sin^4 A - 6 sin^2 A + 2 = 0$

⇒ $sin^4 A - 3 sin^2$ A + 1 = 0

This is quadratic equation in $sin^2$ A

$(sin^2 A)^2 - 3 (sin^2 A) + 1 = 0$

$sin^2 A = {3 ± √{(- 3)^2 - 4 (1) (1)}}/2$

= ${3 ± √5}/2$

$sin^2 A = {3 + √5}/2$ not possible because $sin^2 A$ ≯ 1

So $sin^2 A = {3 - √5}/2$

So none of the options are correct.

Question : 3

If sin θ + cos θ = 1, then what is the value of sin θ cos θ ?

a) 0

b) 2

c) 1

d) $1/2$

Answer: (a)

Given, $\text"sin θ + cos θ"$ = 1

Squaring both sides,

$(sin^2 θ + cos^2 θ) + 2 \text"sin θ cos θ" = 1$

⇒ 1 + 2 sin θ cos θ = 1 ⇒ sin θ cos θ = 0.

Question : 4

If 0 ≤ θ ≤ ${π}/2$ and cos θ + $√3$ sin θ = 2, then what is the value of θ ?

a) ${π}/4$

b) ${π}/3$

c) ${π}/6$

d) ${π}/2$

Answer: (b)

Given that, cos θ + $√3$ sin θ = 2

⇒ $1/2 cos θ + {√3}/2$ sin θ = 1

⇒ sin 30° cos θ + cos 30° sin θ = 1

⇒ sin(30° + θ) = sin 90°

30° + θ = 90°

∴ θ = 60°

Question : 5

The value of cos 25° – sin 25° is

a) positive but greater than 1

b) positive but less than 1

c) negative

d) 0

Answer: (b)

Since, value of cos θ decreases, from 0° to 90° and at 45°, it is equal to the value of sin θ.

Similarly, value of sin θ increases from 0° to 90° and at 45°, it is equal to the value of cos θ

For 0° < 45°, cos θ > sin θ

So, value of cos 25° is always positive but less than 1.

IMPORTANT QUANTITATIVE APTITUDE EXERCISES

Trigonometric Ratios & Identity Model Questions Set 1 Online Quiz

Click to Start..

Recently Added Subject & Categories For All Competitive Exams

Top IBPS PO MAINS 2025: Time Distance Aptitude MCQ PDF

Get Free PDF of Time Distance Aptitude Questions Answer with practice Mock Test and explore various examples for IBPS PO MAINS 2025, Bank, Government exams

13-Feb-2025 by Careericons

Continue Reading »

Top 100+ IBPS PO MAINS 2025: Time Work Aptitude MCQ PDF

Get Free PDF of Time Work Aptitude Questions Answer with a practice Mock Test and explore various examples for IBPS PO MAINS 2025, Bank, Government exams

11-Feb-2025 by Careericons

Continue Reading »

100+ IBPS PO MAINS 2025: Input-Output Reasoning MCQ PDF

Get Free PDF of Input-Output Reasoning Questions Answer with practice Mock Test and explore various examples for IBPS PO MAINS 2025, Bank, Government exams

10-Feb-2025 by Careericons

Continue Reading »

IBPS PO MAINS 2025: Sentence Improvement Questions PDF

Get Free PDF of Sentence Improvement English Questions Answer with practice Mock Test and explore various examples for IBPS PO MAINS 2025, Bank, Govt Exams

07-Feb-2025 by Careericons

Continue Reading »