Question

In quadrant 1:

SIN U = 5/13. COS V = 3/5

Show work to find SIN (U + V).

A. SIN (U + V) = 8/13
B. SIN (U + V) = 2/13
C. SIN (U + V) = 15/13
D. SIN (U + V) = 1/65

Answer 1

To find sin(U + V), we used the trigonometric identity involving sum of angles for the sine function and found values for cos(U) and sin(V) using the Pythagorean identity. The calculated value for sin(U + V) is 63/65, not matching any options provided.

Step-by-step explanation:

We want to find sin(U + V) given that SIN U = 5/13 and COS V = 3/5. We will use the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

1. Using the identity mentioned, let's write it in terms of U and V: sin(U + V) = sin(U)cos(V) + cos(U)sin(V).
2. We already know sin(U) = 5/13 and cos(V) = 3/5, but we need to find cos(U) and sin(V). Since both angles are in quadrant 1, where sine and cosine are positive, we can find these values using the Pythagorean identity: sin²(U) + cos²(U) = 1 and sin²(V) + cos²(V) = 1.
3. Calculate cos(U): since sin(U) = 5/13, cos(U) = √(1 - sin²(U)) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.
4. Calculate sin(V): since cos(V) = 3/5, sin(V) = √(1 - cos²(V)) = √(1 - (3/5)²) = √(1 - 9/25) = √(16/25) = 4/5.
5. Now plug these values into the identity: sin(U + V) = (5/13)(3/5) + (12/13)(4/5) = 15/65 + 48/65 = 63/65.

The answer is not listed among the options provided in the question. Thus, there is likely a mistake or typo in the question itself. However, if we follow the correct procedure, sin(U + V) gives us a result of 63/65, which does not match any of the given answer choices.