Trigonometry
Questions for SSC/Railway Exams (25 – 02 – 2018)
1. What will be the simplified value of (secA - cosA)² +
(cosecA - sinA)² - (cotA - tanA)² ?
a) 0
b) 1
c) 1.2
Answer: B)
Solution: (secA - cosA)² + (cosecA - sinA)² - (cotA - tanA)²
= (sec²A + cos²A - 2 secA
cosA) + (cosec²A + sin²A - 2 cosecA sinA) - (cot²A + tan²A - 2 cotA tanA)
= (sec²A + cos²A - 2) +
(cosec²A + sin²A - 2) - (cot²A + tan²A - 2)
= sec²A - tan²A + cos²A +
sin²A + cosec²A - cot²A - 2
= 3 – 2 = 1
2. What is the value of (tan ⁴ 60° - sin ⁴ 90°) - 2(tan²45° -
3cos0°)²?
a) 3
b) 2
c) 1
d) 0
Answer: D)
Solution :
(tan ⁴ 60° - sin ⁴ 90°) -
2(tan²45° - 3cos0°)²
= ( √ 3 ⁴ - 1 ⁴ ) - 2(1²
- 3*1)² = (9 - 1) - 2(1 - 3)² = 8 - 2*4 = 0
3. If sec θ + tan θ = 3, what is the value of tan θ ?
a) 4/3
b) 3/2
c) 5/2
d) 5/4
Solution :
We know that sec² θ -
tan² θ = 1.
Applying the formula (a²
- b²) = (a + b)(a - b), we get
(sec θ + tan θ )(sec θ -
tan θ ) = 1
Given, sec θ + tan θ =
3 ...(i)
⇒ 3 * (sec θ - tan θ ) = 1
⇒ (sec θ - tan θ ) = 1/3 ...(ii)
Solving equations (i) and
(ii)
⇒ 2tan θ = 3 - 1/3 ⇒ 2tan θ = 8/3 ⇒ tan θ = 4/3
4. If A and B are acute angles, sin (A - B) = 1/2 and cos (A
+ B) = 1/2, what are are values of A and B respectively?
a) 30°, 30°
b) 45°, 15°
c) 30°, 15°
d) 60°, 30°
Answer: B)
Solution :
sin (A - B) = 1/2
⇒ sin (A - B) = sin 30° ⇒ A - B = 30° …(i)
cos (A + B) = 1/2 ⇒ A + B = 60° …(ii)
From (i) and (ii), A =
45°, B = 15°
5. What is the value of [cosec (90° - θ ) - sin (90° - θ )]
[cosec θ - sin θ ] [tan θ + cot θ ]?
a) 0
b) 1
c) -1
d) 1/2
Answer: B)
Solution :
[cosec (90° - θ ) - sin
(90° - θ )] [cosec θ - sin θ ] [tan θ + cot θ ]
= (sec θ - cos θ ) (cosec
θ - sin θ ) (tan θ + cot θ )
= (1/cos θ - cos θ )
(1/sin θ - sin θ ) (tan θ + 1/tan θ )
= {(1 - cos² θ )/cos θ }
{(1 - sin² θ )/sin θ } {(1 + tan² θ )/tan θ }
= (sin² θ /cos θ ) (cos²
θ /sin θ ) (sec² θ /tan θ ) = 1
6. If sin θ . cos θ = ½, what is the value of (sin θ - cos θ
)?
a) 2
b) 1
c) 0
d) -1
Answer: C)
Solution :
Given, sin θ . Cos θ =
1/2
⇒ 2 . sin θ . cos θ = 1 ⇒ sin 2 θ = sin 90° ⇒ 2 θ = 90° ⇒ θ = 45°
∴ (sin θ - cos θ ) = sin 45° - cos 45° = 0
7. What is the value of (cot ⁴ θ - cosec ⁴ θ + cot² θ +
cosec² θ )?
a) -1
b) 0
c) 1
d) 2
Answer: B)
Solution :
(cot ⁴ θ - cosec ⁴ θ +
cot² θ + cosec² θ )
= (cot² θ - cosec² θ )
(cot² θ + cosec² θ ) + cot² θ + cosec² θ
= -1 * (cot² θ + cosec² θ
) + cot² θ + cosec² θ
= - cot² θ - cosec² θ +
cot² θ + cosec² θ = 0
8. If cos θ = 3/5, what is the value of sin θ . sec θ . tan θ
?
a) 3/4
b) 4/3
c) 9/16
d) 16/9
Answer: D)
Solution :
Given, cos θ = 3/5
⇒ sin θ = 4/5 ⇒ sin θ /cos θ = 4/3
sin θ . sec θ . tan θ =
sin θ . (1/cos θ ) . (sin θ /cos θ ) = (sin θ /cos θ )² = (4/3)² = 16/9
9. If a . sin θ + b . cos θ = c, what is the value of a . cos
θ - b . sin θ ?
a) ± √ (-a² + b² + c²)
b) ± √ (a² - b² - c²)
c) ± √ (a² + b² - c²)
d) ± √ (a² - b² + c²)
Answer: C)
Solution :
Given, a . sin θ + b .
cos θ = c
Let a . cos θ - b . sin θ
= k
Squaring and adding both
equations we get:
a² + b² = c² + k²
⇒ k = ± √ (a² + b² - c²)
10. If P . cos θ + Q . sin θ = 8 and P . sin θ - Q . cos θ =
5, what is the value of (P² + Q²)?
a) 39
b) 40
c) 13
d) 2√10
Answer: A)
Solution :
Given, P . cos θ + Q .
sin θ = 8
⇒ P² . cos² θ + Q² . sin² θ + 2PQ . sin θ . cos θ = 64 … (i)
Also given, P . sin θ - Q
. cos θ = 5
⇒ P² . sin² θ + Q² . cos² θ - 2PQ . sin θ . cos θ =
25 … (ii)
Adding equations (i) and
(ii)
P² + Q² = 89
