Trigonometry Questions with Solution for SSC CGL, MTS, CHSL with PDF at Smartkeeda

Direction: Study the following questions carefully and choose the right answer:
1
The value of  (sin 39°) / (cos 51°)  +  2 tan11° tan31° tan45° tan59° tan79°  –  3 (sin2 21° + sin2 69°) is :
» Explain it
D
(sin 39°) / (cos 51°)  +  2 tan11° tan31° tan45° tan59° tan79°  –  3 (sin2 21° + sin2 69°)

(sin 39°) / (sin 39°)  +  2 tan 11° cot 11° tan 31° cot 31° tan 45° – 3(sin2 21° + cos2 21°)

[∵  cos (90° – Θ) = sin Θ, tan (90° – Θ) = cot Θ   &   sin (90° – Θ) = cos Θ]

= 1 + 2 × 1 × 1 × 1 – 3 × 1

[  ∵  tan Θ cot Θ = 1, tan 45° = 1    &    sin2 Θ + cos2 Θ = 1  ]

= 1 + 2 – 3 = 0

Hence, option D is correct.

2
If  cos2 Θ  /  (cot2 Θ – cos2 Θ)   = 3 and 0° < Θ < 90°, then the value of Θ is :
» Explain it
C
cos2 Θ  = 3
cot2 Θ – cos2 Θ

⇒ cos Θ = 3 cot Θ – 3 cos Θ
 
⇒ 4 cos Θ = 3  cos Θ
sin Θ

⇒ 4 sin Θ = 3

⇒ sin Θ =  3
4

⇒ sin Θ = 
3
 = sin 60°
2

⇒ Θ = 60°

Hence, option C is correct.

3
If A = tan 11° tan 29°, B = 2 cot 61° cot 79°, then :
» Explain it
C
Given, A = tan 11° tan 29°

And, B = 2 cot 61° cot 79°

Or, B = 2 tan 29° tan 11° = 2A

[∵  cot (90° – Θ) = tan Θ]

So, 2A = B.

Hence, option C is correct.

4
If sin 17° = x / y, then the value of (sec 17° – sin 73°) is
» Explain it
B
Given, sin 17° =  x
y

sec 17° – sin 73° =  1  – sin (90° – 17°)
cos 17°

[ ∵ sec Θ =  1 ]
cos Θ

1  – cos 17°
cos 17°

[ ∵ sin (90° – Θ) = cos Θ ]

1 – cos2 17°
cos 17°

sin2 17°
1 – sin2 17°

[∵ 1 – cos2 Θ = sin2 Θ & cos Θ = 1 – sin2 Θ ]

x2
y2
1 –  x2
y2

x2
y2
 = 
x2
y y2 – x2
y– x2
y

Hence, option B is correct.

5
The expression   tan 57° + cot 37°  is equal to 
tan 33° + cot 53°
» Explain it
B
tan 57° + cot 37°
tan 33° + cot 53°

 =  tan (90° – 33°) + cot (90° – 53°)
tan 33° + cot 53°

cot 33° + tan 53°  
tan 33° + cot 53°

[ ∵ tan (90° – Θ) = cot Θ  &  cot (90° – Θ) = tan Θ ]

1  + tan 53°
tan 33°
tan 33° +  1
tan 53°

[ ∵ cot Θ = 
1
tan Θ
]

1 + tan 53° tan 33°
tan 33°
tan 33° tan 53° + 1
tan 53°

tan 53°  = cot 33° tan 53°
tan 33°

[ ∵  1  = cot Θ ]
tan Θ

= cot (90° – 57°) tan (90° – 37°)

= tan 57° cot 37°        

[∵ cot (90° – Θ) = tan Θ  &  tan (90° – Θ) = cot Θ]

Hence, option B is correct.