Find the exact value of the trigonometric function given that
sin u = 5/13 and cos = -3/5
1) sin(u-v)
My teacher said to make two right triangles and then find the missing parts by the pyth theorem.
For the 5/13 triangle I got 12 and the other triangle I got 4. I do not know what to do now.
To find the value of sin(u-v), we need to use trigonometric identities. One of the identities that relates sin(u-v) to sin(u) and sin(v) is:
sin(u-v) = sin(u)cos(v) - cos(u)sin(v)
Given that sin(u) = 5/13 and cos(v) = -3/5, we can substitute these values into the identity:
sin(u-v) = (5/13)(-3/5) - cos(u)sin(v)
Now, we need to find the value of cos(u)sin(v) to complete the calculation. To do this, let's draw the right triangles you mentioned.
For the triangle with sin(u) = 5/13, you found one side to be 12 and the other side to be 4 using the Pythagorean Theorem. Therefore, the value of cos(u) can be determined as the adjacent side divided by the hypotenuse, i.e., cos(u) = 12/13.
For the triangle with cos(v) = -3/5, we can apply the Pythagorean Theorem again since we have one side (the adjacent side) equal to 3, and we need to find the hypotenuse. Using a = 3, b = sqrt(h^2 - 3^2), and c = h, where h is the hypotenuse, we can solve for h:
(3^2) + (sqrt(h^2 - 3^2))^2 = h^2
9 + (h^2 - 9) = h^2
h^2 - h^2 + 9 = 9
0 = 0
This means that any value of h will satisfy the equation, so we can choose h = 1 for simplicity.
Now that we have the values of sin(u), cos(v), and cos(u), we can substitute these into the equation for sin(u-v):
sin(u-v) = (5/13)(-3/5) - (12/13)(1/1)
simplified further:
sin(u-v) = -15/65 - 12/13
Calculate the common denominator and perform the subtraction:
sin(u-v) = (-15 - (12 * 5))/65
sin(u-v) = (-15 - 60)/65
sin(u-v) = -75/65
Simplifying the fraction:
sin(u-v) = -15/13
Therefore, the exact value of sin(u-v) is -15/13.