﻿ Class 11 NCERT Math Solution
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TOPICS
Miscellaneous

Example-1 :-  If sin x = 3/5, cos y = -12/13, where x and y both lie in second quadrant, find the value of sin (x + y).

Solution :-
We know that sin (x + y) = sin x cos y + cos x sin y ... (1)
cos² x = 1 - sin² x
= 1 - (3/5)²
= 1 - 9/25
= 16/25
cos x = ± 4/5
Here, x lies on the second quadrant, so cos x is negative.
cos x = -4/5

Now, sin² y = 1 - cos² y
= 1 - (-12/13)²
= 1 - 144/169
= 25/169
sin y = ± 5/13
Here, y lies on the second quadrant, so sin y is positive.
sin y = 5/13

So, put the put the values of sin x, siny, cos x and cos y in (1)
sin (x + y) = sin x cos y + cos x sin y
= 3/5 x (-12/13) + (-4/5) x 5/13
= -36/65 + (-20/65)
sin (x + y) = -56/65

Example-2 :-  Prove that: cos 2x . cos x/2 - cos 3x . cos 9x/2 = sin 5x . sin 5x/2

Solution :-

Example-3 :-  Find the value of tan π/8.

Solution :-
Let x = π/8, then 2x = π/4
Now, tan 2x = (2tan x)/(1 - tan² x)
tan π/4 = (2tan π/8)/(1 - tan² π/8)
Let y = tan π/8.
Then, 1 = 2y/(1 - y²)
1 - y² = 2y
y² + 2y - 1 = 0
By discrimination rule a = 1, b = 2, c = -1
d = b² - 4ac
= 2² - 4 x 1 x (-1)
= 4 + 4
d = 8
Now, y = (-b ± √d)/2a
= (-2 ± √8)/(2 x 1)
= (-2 ± 2√2)/2
= 2(-1 ± √2)/2
y = -1 ± √2
Therefore, π/8 lies on the first quadrant, y = tan π/8 is positive.
Hence, tan π/8 = -1 + √2 = √2 - 1

Example-4 :-  If tan x = 3/4, π < x < 3π/2, find the value of sin x/2, cos x/2 and tan x/2.

Solution :-
Since π < x < 3π/2 (3rd quadrant), cos x is negative
Also, π/2 < x/2 < 3π/4 (2nd quadrant)
Therefore, sin x/2 is positive and cos x/2 is negative.
Now,
sec² x = 1 + tan² x
= 1 + (3/4)²
= 1 + 9/16
= 25/16
sec x = 5/4
1/cos x = 5/4
cos x = 4/5
Here, x lies on the second quadrant, so cos x is negative.
cos x = -4/5

Now, 2sin² x/2 = 1 - cos x
= 1 - (-4/5)
= 1 + 4/5
= 9/5
sin² x/2 = 9/10
sin x/2 = 3/√10 (π/2 < x/2 < 3π/4, sin x/2 is positive)

Now, cos² x/2 = 1 - sin² x/2
= 1 - (3/√10)²
= 1 - 9/10
cos² x/2 = 1/10
cos x/2 = 1/√10
cos x/2 = -1/√10 (π/2 < x/2 < 3π/4, cos x/2 is negative)

Now, tan x/2 = (sin x/2)/(cos x/2)
= (3/√10)/(-1/√10)
= 3/√10 x (-√10/1)
tan x/2 = -3

Example-5 :-  Prove that: cos² x + cos² (x + π/3) + cos² (x - π/3) = 3/2

Solution :-

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