Question

Find the following. tan , given sin 0, tan 2 (Simplify your answer, including any radicals. Use

Answer 1

Answer 2

SOLUITION :

1.

sin^2 ( θ/2) = ( 1 - cos  θ)/2

cos^2 (θ/2) = (1 + cos θ )/2

 

=> tan (θ/2) = sqrt ((1 - coś  θ)/(1 + coś θ))

Now,

sin θ = 3/5 

=> cos^2 (θ) = 1 - sin^2 (θ) = 1 - (3/5)^2 = 16/25

=> cos θ = sqrt(16/25) = 4/5 (positive value as θ is in first quadrant)

So,

tan (θ/2) = sqrt((1 - 4/5)/(1+4/5)) = sqrt(1/9) 

=>  tan (θ/2) = 1/3 (ANSWER). (positive, since θ/2 will be in first quadrant)

2.

tan^2 (β) = sec^2 (β) - 1 = 1/cos^2 (β) - 1

=> (√5/2)^2 = 1/cos^2 (β) - 1

=> cos^2 (β) = 1 / (5/4 + 1) = 4/9

=> cos β = sqrt (4/9) = +/- 2/3 

β is in 3rd quadrant.

So, cos β = - 2/3 

Now,

tan^2 (β/2) = sin^2 (β/2) / cos^2 (β/2) = (1 - cos β)/(1+ cos β). (from trigonometry identity)

= (1 - (- 2/3)) / (1 + (-2/3))

= (1 + 2/3) / (1 - 2/3)

= (5/3) / (1/3)

= 5 

=> tan (β/2) = - √5 (Since β/2 will be in 2nd quadrant)

Hence, tan (β/2) = - √5 (ANSWER).