Final answer:
To find cos(2q) and sin(2q), use the double-angle formulas sin(2q) = 2sin(q)cos(q) and cos(2q) = 1 - 2sin^2(q) after finding cos(q) from sin(q) using the Pythagorean identity.
Step-by-step explanation:
The question asks to find cos(2q) and sin(2q) given that sin(q) = -2/3. To solve for these, we can use the double-angle formulas for sine and cosine:
- sin(2q) = 2sin(q)cos(q)
- cos(2q) = cos2(q) - sin2(q), which can also be expressed as 2cos2(q) - 1 or 1 - 2sin2(q)
Since we have sin(q), we need to find cos(q). The Pythagorean identity sin2(q) + cos2(q) = 1 can help us, where cos2(q) = 1 - sin2(q). Substituting sin(q) = -2/3, we get cos2(q) = 1 - (-2/3)2 = 1 - 4/9 = 5/9. Therefore, cos(q) can be ±5/3, but we must consider the quadrant in which q lies to determine the sign of cos(q).
To find sin(2q) using the given sin(q), assume cos(q) to be positive and negative in the respective quadrants, apply the double-angle formula: sin(2q) = 2 * -2/3 * ±5/3. Now use the formula for cos(2q), which is 1 - 2sin2(q) = 1 - 2 * (-2/3)2 to get the value of cos(2q).