tan θ = 12/5, π < θ < 3π/2 Find sin (θ/2)?
Given that ( \tan \theta = \frac{12}{5} ) and ( \pi < \theta < \frac{3\pi}{2} ), how can we find ( \sin\left(\frac{\theta}{2}\right) )?
2 Answers
tan θ = 12/5 and θ is in III quadrant where sin and cos are negative
opposite side = 12 and adj. side = 5
hypotenuse = sqrt(144+25) = 13
cos θ = -5/13
==> 1 - 2sin^2(θ/2) = -5/13
2sin^2(θ/2) = 1 + (5/13)
= 18/13
sin^2(θ/2) = 9/13
sin (θ/2) = (3/13)√13
Feb 22, 2025
tan(t) = -3/4 sec(t)^2 = a million + tan(t)^2 sec(t)^2 = a million + 9/sixteen sec(t)^2 = 25/sixteen cos(t)^2 = sixteen/25 cos(t) = 4/5 sin(t/2) => sqrt((a million - cos(t)) / 2) => sqrt((a million - 4/5) / 2) => sqrt(a million/10) => +/- sqrt(10) / 10 on account that t is in this autumn, then t/2 might properly be the two in Q2 or in this autumn, so the two solutions are valid
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