I have these sum and product property problems and I just want to make sure my work is right.
Write equations expressing
1. cos 37degrees cos99degrees as a sum
Answer: sin136+(-62)
2. sin4+sin7 as a product
Answer: 2sin5.5 sin-1.5
3. cos 8x - cos 2x as a product
Answer -2sin5x sin3x
4. cos5x sin9x as a sum
Answer sin14x - (-4)
Idk...I just don't feel as confident about these.
Let's go through each problem and see if your answers are correct. I'll explain the steps to finding the correct answers, and you can check if your work aligns with the explanations.
1. To express cos(37°) * cos(99°) as a sum, we can use the cosine sum formula:
cos(A) * cos(B) = 1/2[cos(A+B) + cos(A-B)]
Plugging in the given angles, we have:
cos(37°) * cos(99°) = 1/2[cos(37° + 99°) + cos(37° - 99°)]
Simplifying further:
cos(37° + 99°) = cos(136°) = sin(180° - 136°) = sin(44°)
cos(37° - 99°) = cos(-62°) = cos(360° - 62°) = cos(298°)
So, the expression as a sum is sin(44°) - cos(298°).
Your answer, sin136 + (-62), is incorrect. It should be sin(44°) - cos(298°).
2. To express sin(4) + sin(7) as a product, we can use the product-to-sum identities:
sin(A) + sin(B) = 2sin((A + B) / 2) * cos((A - B) / 2)
Plugging in the given angles, we have:
sin(4) + sin(7) = 2sin((4 + 7) / 2) * cos((4 - 7) / 2)
Simplifying further:
sin((4 + 7) / 2) = sin(11 / 2) = sin(5.5)
cos((4 - 7) / 2) = cos(-3 / 2) = cos(3/2)
So, the expression as a product is 2sin(5.5) * cos(3/2).
Your answer, 2sin5.5 sin-1.5, is incorrect. It should be 2sin(5.5) * cos(3/2).
3. To express cos(8x) - cos(2x) as a product, we can use the cosine difference formula:
cos(A) - cos(B) = -2sin((A + B)/2) * sin((A - B)/2)
Plugging in the given angles, we have:
cos(8x) - cos(2x) = -2sin((8x + 2x)/2) * sin((8x - 2x)/2)
Simplifying further:
sin((8x + 2x)/2) = sin(10x/2) = sin(5x)
sin((8x - 2x)/2) = sin(6x/2) = sin(3x)
So, the expression as a product is -2sin(5x) * sin(3x).
Your answer, -2sin5x sin3x, is correct.
4. To express cos(5x) * sin(9x) as a sum, we can use the sine product-to-sum identity:
cos(A) * sin(B) = 1/2[sin(A + B) + sin(A - B)]
Plugging in the given angles, we have:
cos(5x) * sin(9x) = 1/2[sin(5x + 9x) + sin(5x - 9x)]
Simplifying further:
sin(5x + 9x) = sin(14x)
sin(5x - 9x) = sin(-4x) = -sin(4x)
So, the expression as a sum is sin(14x) - (-sin(4x)), which is equivalent to sin(14x) + sin(4x).
Your answer, sin14x - (-4), is incorrect. It should be sin(14x) + sin(4x).
Overall, you had some incorrect answers in questions 1 and 2, while questions 3 and 4 had one correct answer and one incorrect answer. I hope this explanation helps you understand the steps involved in finding the right answers.