If sin(x) = 3/5 and x is in the second quadrant (pi/2 < x < pi), what is the exact value of tan(2*x)?
Answer and explanation
Correct answer: D. -24/7
- Step 1: In quadrant II, cos(x) = -sqrt(1 - (3/5)^2) = -4/5.
- Step 2: Compute tan(x) = sin(x)/cos(x) = (3/5)/(-4/5) = -3/4.
- Step 3: Apply double angle formula: tan(2x) = 2tan(x) / (1 - tan(x)^2) = 2*(-3/4) / (1 - 9/16) = (-3/2) / (7/16) = -24/7.
Answer
-24/7.
Why not the others: 24/7 has the wrong sign due to neglecting that x is in quadrant II; -7/24 inverts numerator and denominator; 7/25 is cos(2x), not tan(2x).
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