heya!!
Derive function of 120 degree from the functions of 60 degree and check by using relation between functions of supplementary angles.
thanks :D
Answers
Step-by-step explanation:
Given, θ = 60°.
Now, we have to derive functions of 120 degrees(2 * 60 = 2θ)
(i)
sin 2θ = 2 sinθ cosθ
= 2 sin60 cos60
= 2 (√3/2) (1/2)
= √3/2
Verification:
Supplementary angle = 180°.
sin 120 = sin(180 - 60)
= sin 60 {sin (180 - θ) = sin θ}
= √3/2
∴ Hence, verified!
(ii)
cos 2θ = 1 - 2sin²θ
= 1 - 2(3/4)
= -1/2
Verification:
cos 120 = cos(180 - 60)
= -cos 60 {cos(180 - θ) = -cosθ}
= -1/2
∴Hence, verified!
(iii)
tan 2θ = sin 2θ/cos 2θ
= (√3/2) * (-2)
= -√3
Verification:
tan 120 = tan(180 - 60)
= tan 60 {tan(180 - θ) = -tanθ}
= -√3
∴ Hence, verified!
(iv)
cot 2θ = cos 2θ/sin 2θ
= (-1/2) * (2/√3)
= -1/√3
Verification:
cot 120 = cot(180 - 60)
= -cot 60 {cot (180 - 60} = -cot 60}
= -1/√3
∴Hence, verified!
(v)
sec 2θ = (1/cos2θ)
= (1/-1/2)
= -2
Verification:
sec 120 = sec(180 - 60)
= -sec 60 {sec (180 - θ) = -secθ}
= -2.
Hence, verified!
(vi)
cosec 2θ = (1/sin 2θ)
= (1/√3/2)
= 2/√3
Verification:
cosec 120 = cosec (180 - 60)
= cosec 60 {cosec(180 - θ) = cosecθ}
= 2/√3
∴ Hence, verified!
Hope it helps!