Question

Help meeeeeeeeeeeeeeeeeeee

Answer 1

Answer:

  -336/527

Step-by-step explanation:

Trig identities are involved.

  tan(2x) = 2tan(x)/(1 -tan(x)²)

Using tan = sin/cos, we can write this in terms of sine and cosine as ...

  tan(2x) = (2sin(x)/cos(x))/(1 -sin(x)²/cos(x)²) = 2sin(x)cos(x)/(cos(x)² -sin(x)²)

  = 2sin(x)cos(x)/(1 -2sin(x)²)

Now, the cosine can be found from ...

  cos(x) = √(1 -sin(x)²)

for sin(θ) = 24/25, cos(θ) = √(1 -(24/25)²) = 7/25 . . . . 1st quadrant angle

Filling in the values in the above identity, we have ...

  tan(2θ) = 2(24/25)(7/25)/(1 -2(24/25)²) = 336/-527

  tan(2θ) = -336/527

_____

You can use a calculator or estimate that the angle for sin(θ) = 24/25 will be greater than 67.5°, so double the angle will be greater than 135°. (θ ≈ 74°) This means ...

• 2θ > 135°, so the magnitude of tan(2θ) will be less than 1
• 2θ is in the 2nd quadrant, so the sign of tan(2θ) will be negative

These observations will help you choose the correct answer without any further math.