If sin(x) = /45 and cos(y) = 5/13 with both x and y terminating in quadrant 1 find the exact value of cos(x-y)
cos(4/5 - 5/13)
Is this what I would do?
Your sin(x) fraction lacks a numerator
the sine is 4/5 sorry!
No, that is not what you would do. Call the angles A and B.
For the angles you have chosen,
cos(x) = 3/5 and sin(y) = 12/13. You can prove that with the Pythagorean theorem, or sin^2 + cos^2 = 1. That will make the calculation easier.
Then use the identity:
cos(A-B) = cosA cosB + sinA sin B
= (3/5)(5/13) + (4/5)(12/13)
= (15 + 48)/65 = 63/65
To find the exact value of cos(x-y), you need to use the trigonometric identity: cos(x-y) = cos(x)cos(y) + sin(x)sin(y).
In this case, sin(x) = √45 and cos(y) = 5/13.
First, let's find out the value of sin(y). Since y terminates in quadrant 1, sin(y) will be positive. Using the Pythagorean identity, sin^2(y) + cos^2(y) = 1, we can solve for sin(y).
cos^2(y) = 1 - sin^2(y) = 1 - (5/13)^2 = 1 - 25/169 = 144/169
Taking the square root of both sides, sin(y) = √(144/169) = 12/13
Now, we have all the values we need to calculate cos(x-y).
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
= (√45)(5/13) + (12/13)(√45)
Simplifying the expression,
cos(x-y) = (5√45)/13 + (12√45)/13
= (5√45 + 12√45)/13
Finally, we can simplify this expression further by combining like terms under the square root:
cos(x-y) = (√45(5 + 12))/13
= (√45(17))/13
= (3√5(17))/13
So the exact value of cos(x-y) is (3√5(17))/13.