Answer:
To find the exact value of the trigonometric function 4cos(a)sin(B) in two ways, we can use trigonometric identities and the given figure.
First way:
We know that the trigonometric identity for the product of cosine and sine is:
cos(a)sin(B) = (1/2)[sin(a + B) - sin(a - B)]
Using this identity, we can rewrite the expression:
4cos(a)sin(B) = 4[(1/2)[sin(a + B) - sin(a - B)]]
Simplifying further:
4cos(a)sin(B) = 2[sin(a + B) - sin(a - B)]
Second way:
In the given figure, we can see that angle a and angle B are complementary angles. This means that their sum is equal to 90 degrees or π/2 radians.
Using this information, we can rewrite the expression using trigonometric identities:
4cos(a)sin(B) = 4cos(a)cos(π/2 - B)
We can also use the identity for the product of cosine and cosine:
cos(a)cos(π/2 - B) = (1/2)[cos(a + (π/2 - B)) + cos(a - (π/2 - B))]
Simplifying further:
4cos(a)sin(B) = 2[cos(a + (π/2 - B)) + cos(a - (π/2 - B))]
In both ways, we have found the exact value of the trigonometric function 4cos(a)sin(B) using trigonometric identities and the given figure.