If sinx=53,cosy=−1312, where x and y both lie in second quadrant, find the value of sin(x+y).
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Step-by-Step Solution
Step 1: Find cos x and sin y
Since x and y both lie in the second quadrant, cosx will be negative and siny will be positive. We use the Pythagorean identity sin2θ+cos2θ=1 to find the missing trigonometric values.
Step 2: Calculate cos x
Substitute the given value of sinx=53 into the formula for cosx. Simplify the expression under the square root to find the value of cosx.
Step 3: Calculate sin y
Substitute the given value of cosy=−1312 into the formula for siny. Simplify the expression under the square root to find the value of siny.
Step 4: Apply the sum formula for sine
Now that we have all the necessary trigonometric values, we can use the sum formula for sine to find sin(x+y).
Step 5: Substitute values and calculate
Substitute the calculated values of sinx, cosy, cosx, and siny into the sum formula. Perform the multiplication and then add the resulting fractions to get the final answer.