If sin(θ)=−24/25, and θ is in Quadrant III, then what is tan(θ/2)?

Answer 1

To solve this problem, we can use the formula:

tan(θ/2) = sin (θ) /(1 + cos(θ))

We are given that sin(θ) = -24/25 and the angle is in quadrant III, so if we draw our right triangle in the coordinate plane and do the pythagorean theorem:

x² + (- 24)² = 25² to find the x-side of the triangle, we get

x² =49 which in turn gives us x = - 7 (since we're in quadrant III)

Now, we can figure out cos(θ) which is:

cos(θ) = -7/25

So we can plug it into our formula:

tan(θ/2) = sin (θ) /(1 + cos(θ)) which gives us:

(-24/25)/(1 + (-7/25))

= (-24/25)/((25/25) + (-7/25))

= (-24/25)/(18/25)

We then flip the fraction in the denominator to get

= (-24/25) * (25/18)

= -24/18

which simplifies to

= - 4/3