Connect with experts and get insightful answers on IDNLearn.com. Join our interactive community and get comprehensive, reliable answers to all your questions.
Sagot :
Certainly! Let's write the given product [tex]\( 12 \cos (8q) \sin (7q) \)[/tex] as a sum using trigonometric identities. We will use the product-to-sum formulas, specifically, the identity:
[tex]\[ \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \][/tex]
Here, [tex]\( A = 7q \)[/tex] and [tex]\( B = 8q \)[/tex]. Therefore:
[tex]\[ \cos(8q) \sin(7q) = \frac{1}{2}[\sin(7q + 8q) + \sin(7q - 8q)] = \frac{1}{2}[\sin(15q) + \sin(-q)] \][/tex]
Since [tex]\( \sin(-x) = -\sin(x) \)[/tex], we can rewrite [tex]\( \sin(-q) \)[/tex]:
[tex]\[ \sin(-q) = -\sin(q) \][/tex]
Now, substituting this back into our formula, we get:
[tex]\[ \cos(8q) \sin(7q) = \frac{1}{2}[\sin(15q) - \sin(q)] \][/tex]
We need to multiply this expression by 12 to get the original expression:
[tex]\[ 12 \cos(8q) \sin(7q) = 12 \cdot \frac{1}{2}[\sin(15q) - \sin(q)] \][/tex]
[tex]\[ = 6[\sin(15q) - \sin(q)] \][/tex]
[tex]\[ = 6\sin(15q) - 6\sin(q) \][/tex]
Thus, the expression [tex]\( 12 \cos(8q) \sin(7q) \)[/tex] written as a sum is:
[tex]\[ 12 \cos(8q) \sin(7q) = 6\sin(15q) - 6\sin(q) \][/tex]
This is your final answer.
[tex]\[ \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \][/tex]
Here, [tex]\( A = 7q \)[/tex] and [tex]\( B = 8q \)[/tex]. Therefore:
[tex]\[ \cos(8q) \sin(7q) = \frac{1}{2}[\sin(7q + 8q) + \sin(7q - 8q)] = \frac{1}{2}[\sin(15q) + \sin(-q)] \][/tex]
Since [tex]\( \sin(-x) = -\sin(x) \)[/tex], we can rewrite [tex]\( \sin(-q) \)[/tex]:
[tex]\[ \sin(-q) = -\sin(q) \][/tex]
Now, substituting this back into our formula, we get:
[tex]\[ \cos(8q) \sin(7q) = \frac{1}{2}[\sin(15q) - \sin(q)] \][/tex]
We need to multiply this expression by 12 to get the original expression:
[tex]\[ 12 \cos(8q) \sin(7q) = 12 \cdot \frac{1}{2}[\sin(15q) - \sin(q)] \][/tex]
[tex]\[ = 6[\sin(15q) - \sin(q)] \][/tex]
[tex]\[ = 6\sin(15q) - 6\sin(q) \][/tex]
Thus, the expression [tex]\( 12 \cos(8q) \sin(7q) \)[/tex] written as a sum is:
[tex]\[ 12 \cos(8q) \sin(7q) = 6\sin(15q) - 6\sin(q) \][/tex]
This is your final answer.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.