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 :
- ${cos 75°}/{sin 15 °} + {sin 12°}/{cos 78°} - {cos 18°}/{sin 72°}$ = 1
- ${cos 35°}/{sin 55°} - {sin 11°}/{cos 79°}$ + cos 28° cosec 62°= l
- ${sin 80°}/{cos 10°}$ - sin 59° sec 31° = 0
a) 2 and 3 only
b) 1 and 2 only
c) 1 and 3 only
d) 1, 2 and 3
Answer »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 »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 »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 »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 »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.
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