NCERT Solutions Class 11 Mathematics Trigonometric-Functions Exercise 3.2

Question 1:
Find the values of other five trigonometric functions if cos x = -1/2, x lies in third quadrant.

Answer:
Given, cos x = -1/2
So, sec x = 1/cos x = 1/(-1/2) = -2
We know that
      sin² x + cos² x = 1
⇒ sin² x =1 – cos² x
⇒ sin² x =1 – (-1/2)²
⇒ sin² x = 1 – 1/4
⇒ sin² x = 3/4
⇒ sin x = ±√3/2
Since x lies in the 3^rd quadrant, the value of sin x will be negative.
So, sin x = -√3/2
cosec x = 1/sin x = 1/(-√3/2) = -2/√3
tan x = sin x/cos x = (-√3/2)/ (-1/2) = √3
cot x = 1/1tan x = 1√3

Question 2:
Find the values of other five trigonometric functions if sin x = 3/5, x lies in second quadrant.

Answer:
Given, sin x = 3/5
cosec x = 1/ sin x = 1/(3/5) = 5/3
We know that
      sin² x + cos² x = 1
⇒ cos² x =1 – sin ² x
⇒ cos² x =1 – (3/5)²
⇒ cos² x = 1 – 9/25
⇒ cos² x = 16/25
⇒ cos x = ±4/5
Since x lies in the 2^nd quadrant, the value of cos x will be negative.
So, sin x = -4/5
sec x = 1/cos x = 1/(-4/5) = -5/4
tan x = sin x/cos x = (3/5)/(-4/5) = -2/4
cot x = 1/tan x = 1/(-3/4) = -4/5

Question 3:
Find the values of other five trigonometric functions if cot x = 3/4, x lies in third quadrant.

Answer:
Given, cot x = 3/4
tan x = 1/cot x = 1/(3/4) = 4/3
We know that
      1 + tan² x = sec² x
⇒ 1 + (4/3)²  = sec² x
⇒ sec² x = 1 + 16/9
⇒ sec² x = 25/9
⇒ sec x = ±5/3
Since x lies in the 3^rd quadrant, the value of sec x will be negative.
So, sec x = -5/3
cos x = 1/sec x = 1/(-5/3) = -3/5
Again, tan x = sin x/cos x
⇒ 4/3 = sin x/(-3/5)
⇒ sin x = (4/3) * (-3/5)
⇒ sin x = -4/5
cosec x = 1/sin x = 1/(-4/5) = -5/4

Question 4:
Find the values of other five trigonometric functions if sec x = 13/5, x lies in fourth quadrant.

Answer:
Given, sec x = 13/5
So, cos x = 1/1sec x = 1/(13/5) = 5/13
We know that
      sin² x + cos² x = 1  ⇒ sin² x =1 – cos² x
⇒ sin² x =1 – (5/13)²
⇒ sin² x = 1 – 25/169
⇒ sin² x = 144/169
⇒ sin x = ±12/13
Since x lies in the 4^th quadrant, the value of sin x will be negative. So, sin x = -12/13
cosec x = 1/sin x = 1/(-12/13) = -13/12
Now, tan x = sin x/cos x = (-12/13)/(5/13) = -12/5
cot x = 1/tan x = 1/(-12/5) = -5/12

Question 5:
Find the values of other five trigonometric functions if tan x = -5/12, x lies in second quadrant.

Answer:
Given, tan x = -5/12
So, cot x = 1/tan x = 1/(-5/12) = -12/5
We know that
      1 + tan² x = sec² x
⇒ 1 + (-5/12)²  = sec² x
⇒ sec² x = 1 + 25/144
⇒ sec² x = 169/12
⇒ sec x = ±13/12
Since x lies in the 2^nd quadrant, the value of sec x will be negative.
So, sec x = -13/12
cos x = 1/sec x = 1/(-13/12) = -12/13
Again, tan x = sin x/cos x
⇒ -5/12 = sin x/(-12/13)
⇒ sin x = (-5/12) * (-12/13)
⇒ sin x = 5/13
cosec x = 1/sin x = 1/(5/13) = 13/5