If x and y are in the interval (0,π/12) and sin x = 3/5 and cos y = 12/13, evaluate:
sin (x-y)
I started by trying to find the value of x and y and i got 0.645 and 0.394. And I substituted the values in and got 0.2461 etc. But the answer at the back of the book says 16/65. Which is 0.2461 etc however I am supposed to find it in a fraction. How can I do that?
You can't just plug those numbers in. You must use the Sum and Difference Identities. It will be helpful to memorize the equation:
sin(A-B) = sinA * cos B - cos A * sin B
You are given two of the numbers you need. Use the #1 Identity, sin^2(x) + cos^(2)x = 1, to find sin y and cos x. Then, just plug & chug.
find the length of each arc of 12 yards and 260 degrees
To evaluate sin(x-y) and express the answer as a fraction, we need to use trigonometric identities and properties.
Firstly, we know that sin(x-y) can be expressed as sin(x)cos(y) - cos(x)sin(y).
Given that sin(x) = 3/5 and cos(y) = 12/13, we can substitute these values into the formula:
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)
= (3/5)(12/13) - cos(x)sin(y)
Now, to determine the value of cos(x)sin(y), we need to find cos(x) and sin(y).
Since x is in the interval (0, π/12), we can use the Pythagorean identity for sin^2(x) + cos^2(x) = 1 to find cos(x):
cos^2(x) = 1 - sin^2(x)
= 1 - (3/5)^2
= 1 - 9/25
= 16/25
cos(x) = ±√(16/25)
= ±4/5
Since y is in the interval (0, π/12), we can use the Pythagorean identity for sin^2(y) + cos^2(y) = 1 to find sin(y):
sin^2(y) = 1 - cos^2(y)
= 1 - (12/13)^2
= 1 - 144/169
= 25/169
sin(y) = ±√(25/169)
= ±5/13
Now, we substitute the values of cos(x) and sin(y) into the equation:
sin(x-y) = (3/5)(12/13) - (4/5)(5/13)
= 36/65 - 20/65
= 16/65
Therefore, sin(x-y) evaluates to 16/65, which matches the answer in the book.