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?
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To find the exact value of cos(x-y), where x and y are given as sin(x) = √(45) and cos(y) = 5/13, both terminating in quadrant 1, you need to follow these steps:
Step 1: Find the values of sin(x) and cos(y).
Given sin(x) = √(45) and cos(y) = 5/13.
Step 2: Recall the trigonometric identity cos(x-y) = cos(x)cos(y) + sin(x)sin(y).
In this case, we need to find cos(x-y), so we will substitute the given values of sin(x) and cos(y) into the identity.
Step 3: Substitute the values into the identity.
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
cos(x-y) = √(45) * (5/13) + √(45) * (5/13)
cos(x-y) = (5√45)/(13) + (5√45)/(13)
cos(x-y) = (10√45)/(13)
Step 4: Simplify the result if possible.
To simplify (√45) and (√45) we can factor out the square root of 9 from 45.
Since 45 = 9 * 5, we have:
cos(x-y) = (10 * √(9 * 5))/(13)
cos(x-y) = (10 * √9 * √5)/(13)
cos(x-y) = (10 * 3 * √5)/(13)
cos(x-y) = (30√5)/(13)
Therefore, the exact value of cos(x-y) is (30√5)/(13).