Misc 8
Find the value of sin 𝑥/2 , cos 𝑥/2 and tan 𝑥/2 in each of the following :
tan𝑥 = – 4/3 , 𝑥 in quadrant II
Given that x is in quadrant II
So,
90° < x < 180°
Dividing by 2 all sides
(90°)/2 < 𝑥/2 < (180°)/2
45° < 𝑥/2 < 90°
So, 𝑥/2 lies in Ist quadrant
In 1st quadrant,
sin , cos & tan are positive
∴ sin 𝑥/2 , cos 𝑥/2 and tan 𝑥/2 are positive
Given
tan x = (−4)/3
We know that
tan 2x = (2 𝑡𝑎𝑛𝑥)/(1 − 𝑡𝑎𝑛2𝑥)
Replacing x with 𝑥/2
tan (2𝑥/2) = (2 𝑡𝑎𝑛(𝑥/2))/(1 − 𝑡𝑎𝑛2(𝑥/2) )
tan x = (2 𝑡𝑎𝑛(𝑥/2))/(1 − 𝑡𝑎𝑛2(𝑥/2) )
(2 tan(𝑥/2))/(1 − 𝑡𝑎𝑛2(𝑥/2) ) = −4/3
−4/3 = (2 tan(𝑥/2))/(1 − 𝑡𝑎𝑛2(𝑥/2) )
–4(2𝑥/2) = 3× 2 tan (𝑥/2)
–4 × 1 – (–4) × tan2 (𝑥/2) = 6 tan (𝑥/2)
–4 × 1 – (–4) × tan2 (𝑥/2) = 6 tan (𝑥/2)
–4 + 4 tan2 (𝑥/2) = 6 tan (𝑥/2)
–4 + 4 tan2 (𝑥/2) – 6 tan (𝑥/2) = 0
Replacing tan 𝒙/𝟐 by a
Our equation becomes
–4 + 4a2 – 6a = 0
4a2 – 6a – 4 = 0
4a2 – 8a + 2a – 4 = 0
4a(a – 2) + 2 (a – 2) = 0
(4a + 2) (a – 2) = 0
Hence
4a + 2 = 0
4a = −2
a = (−2)/( 4)
a = (−1)/2
So, a = (−1)/2 or a = 2
Hence,
tan 𝑥/2 = (−1)/2 or tan 𝑥/2 = 2
Since, 𝑥/2 lies in Ist quadrant
tan 𝑥/2 is positive,
∴ tan 𝒙/𝟐 = 2
Now,
We know that
1 + tan2 x = sec2 x
Replacing x with 𝑥/2
1 + tan2 𝑥/2 = sec2 𝑥/2
1 + (2)2 = sec2 𝑥/2
1 + 4 = sec2 x/2
1 + 4 = sec2 x/2
5 = sec2 𝑥/2
sec2 𝑥/2 = 5
sec 𝑥/2 = ± √5
Since 𝑥/2 lie on the 1st Quadrant,
sec 𝑥/2 is positive in the 1st Quadrant
So, sec 𝒙/𝟐 = √𝟓
Therefore,
cos 𝒙/𝟐 = 𝟏/√𝟓
Now,
We know that
sin2x + cos2x = 1
Replacing x with 𝑥/2
sin2 𝑥/2 + cos2 𝑥/2 = 1
sin2 𝑥/2 = 1 – cos2 𝑥/2
Putting cos 𝑥/2 = √5/5
sin2 𝑥/2 = 1 – (√5/5)2
sin2 𝑥/2 = 1 – 5/25
sin2 𝑥/2 = 1 – 1/5
sin2 𝑥/2 = (5 − 1)/5
sin2 𝑥/2 = 4/5
sin 𝑥/2 = ± √(4/5)
sin 𝑥/2 = ± √4/√5
sin 𝑥/2 = ± 2/√5
sin 𝑥/2 = ± 2/√5 × √5/√5
sin 𝑥/2 = ± (2√5)/5
Since 𝑥/2 lies on the 1st Quadrant
sin 𝑥/2 is positive in the 1st Quadrant
So, sin 𝒙/𝟐 = (𝟐√𝟓)/𝟓
Therefore,
tan 𝑥/2 = 2 , cos 𝒙/𝟐 = √𝟓/𝟓 & sin 𝒙/𝟐 = (𝟐√𝟓)/𝟓

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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