Final answer:
The value of sin(2theta) when sin(theta) = 4/5 is found using the double-angle formula. After computing cos(theta) using the Pythagorean identity, we get sin(2theta) = 24/25, considering the positive value of cos(theta). The correct answer is A.
Step-by-step explanation:
If sin(theta) = 4/5, we want to find sin(2theta). To solve this, we can use the double-angle formula for sine, which states that sin(2theta) = 2 * sin(theta) * cos(theta). Since we already know sin(theta), we first need to find cos(theta). We can do this using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1.
Given sin(theta) = 4/5, squaring both sides gives us sin^2(theta) = 16/25. Substituting into the identity, we have 16/25 + cos^2(theta) = 1. Solving for cos^2(theta), we get cos^2(theta) = 9/25, therefore cos(theta) can be +3/5 or -3/5. Considering the ambiguity of theta's quadrant, we will use both values to find potential answers for sin(2theta).
For cos(theta) = 3/5, sin(2theta) = 2 * 4/5 * 3/5 = 24/25. If cos(theta) = -3/5, sin(2theta) = 2 * 4/5 * -3/5 = -24/25. The sign of sin(2theta) will depend on the specific quadrant in which theta resides. Since no information about theta's quadrant is provided, we typically consider the positive root. Therefore, the answer would most likely be sin(2theta) = 24/25, which is option a).