Final answer:
In trigonometry, given sin(u) and the quadrant where u lies, other trigonometric values can be determined using Pythagorean identities and known periodic properties. In this case, sin(u) = 3/5 and cos(u) is negative imply that u is in the second quadrant, leading to specific values for cos(u), tan(u), and their negatives.
Step-by-step explanation:
The question involves trigonometric values and identities in mathematics. Given that sin(u) = 3/5 and cos(u) is negative, we can determine the other trigonometric values for u. Since sin(u) is positive and cos(u) is negative, u is in the second quadrant where sine is positive and cosine and tangent are negative.
To find cos(u), we use the Pythagorean identity: sin²(u) + cos²(u) = 1. Since sin(u) = 3/5, we have (3/5)² + cos²(u) = 1, leading to cos(u) = -4/5.
To find tan(u), we use tan(u) = sin(u)/cos(u) = (3/5)/(−4/5) = −3/4.
Now, sin(−u) is the negative of sin(u), so sin(-u) = −3/5, and cos(−u) is the same as cos(u), so cos(-u) = −4/5. Similarly, tan(−u) = −tan(u) = 3/4.
For sin(u + π) = sin(u + 7), since 7 is a little more than 2π, which is a full cycle, the sine value would be the same as sin(u). And, for cos(u + π) and tan(u + π), we remember that adding π (or a full cycle plus π) will reverse the sign of the sine and cosine but not affect their absolute values, so cos(u + π) = 4/5 and tan(u + π) = 3/4.
Since trigonometric functions are periodic, we might sometimes use the identities to simplify expressions with added multiples of π or 2π. However, if x is not given as a specific value or multiple of π, precise values cannot be determined.