Use the sum-to-product formula to simplify the expression:
If sin 52 + sin 8 = sin A, 0<A<90, then
A =??? degrees.
sin 52 + sin 8 = 2 sin 30 * cos 22
so
sin A = 2*.5*.927 = .927
A= 68 degrees
To solve this problem, we need to use the sum-to-product formula for trigonometric functions. The sum-to-product formula states:
sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
In this problem, we are given sin(52) + sin(8) = sin(A), and we need to find the value of A.
Using the sum-to-product formula, we can write:
sin(A) = sin(52) + sin(8)
sin(A) = 2sin((52+8)/2)cos((52-8)/2)
sin(A) = 2sin(30)cos(22)
sin(A) = 2(1/2)(√3/2)
sin(A) = √3/2
Now, we know that the sine function is positive in the first and second quadrants. Since 0 < A < 90, A must lie in the first quadrant.
In the first quadrant, the sine function is equal to the opposite side divided by the hypotenuse in a right triangle.
So, we can write:
sin(A) = √3/2 = opposite side / hypotenuse
Since the hypotenuse is always greater than the opposite side, we can conclude that A must be 60 degrees.
Therefore, A = 60 degrees.