use the sum to product formula to simplify
sin 42degrees+sin 18 degrees=sin A degrees 0<A<90 degrees
A=???degrees
Using the usual formulas,
sin42°+sin18° = 2 sin30° cos12°
since sin30° = 1/2, that is just
cos12° = sin78°
cosine means sine of the complement.
To simplify the expression sin 42 degrees + sin 18 degrees, we can use the sum to product formula. The sum to product formula for sine states that:
sin A + sin B = 2 * sin((A + B) / 2) * cos((A - B) / 2)
Using this formula, we can rewrite the expression as:
sin 42 degrees + sin 18 degrees = 2 * sin((42 + 18) / 2) * cos((42 - 18) / 2)
Simplifying further:
sin 42 degrees + sin 18 degrees = 2 * sin(60 / 2) * cos(24 / 2)
sin 42 degrees + sin 18 degrees = 2 * sin(30) * cos(12)
Next, we can use the values of sine and cosine for 30 degrees and 12 degrees respectively. sin 30 degrees equals 1/2 and cos 12 degrees equals approximately 0.9781.
sin 42 degrees + sin 18 degrees = 2 * (1/2) * 0.9781
Finally, simplifying the expression:
sin 42 degrees + sin 18 degrees = 0.9781
Therefore, sin A degrees for 0 < A < 90 degrees is approximately 0.9781.
To simplify the given expression sin 42 degrees + sin 18 degrees = sin A degrees, we can use the sum-to-product formula for sine.
The sum-to-product formula states that sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2).
In this case, A = 42 degrees and B = 18 degrees. Plugging these values into the formula, we get:
sin 42 degrees + sin 18 degrees = 2*sin((42+18)/2)*cos((42-18)/2)
Simplifying further:
sin 42 degrees + sin 18 degrees = 2*sin(60/2)*cos(24/2)
sin 42 degrees + sin 18 degrees = 2*sin(30)*cos(12)
Now, we know that sin(30) is equal to 1/2 and cos(12) is a constant value.
sin 42 degrees + sin 18 degrees = 2 * (1/2) * cos(12)
sin 42 degrees + sin 18 degrees = cos(12)
Therefore, A = 12 degrees.