How do i give the exact value for:
(sin pi/8)(cos pi/24) + (cos pi/8)(sin pi/24)
recall that
sin(A+B) = sinAcosB + cosAsinB
you have the same angle pair showing up in the two terms in that pattern, so
(sin pi/8)(cos pi/24) + (cos pi/8)(sin pi/24)
= sin(pi/8 + pi/24)
= sin(pi/6)
= 1/2 ----> you should know pi/6 =30 degrees and sin(30) = 1/2
To find the exact value of the expression (sin(pi/8))(cos(pi/24)) + (cos(pi/8))(sin(pi/24)), we can use the trigonometric identities for the sum and difference of angles.
Step 1: Identify the trigonometric identities
The identities we need are:
1. sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
2. cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Step 2: Apply the identities
Let A = pi/8 and B = pi/24.
From the first identity:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(pi/8 + pi/24) = sin(pi/8)cos(pi/24) + cos(pi/8)sin(pi/24)
Step 3: Simplify the expression
pi/8 + pi/24 = (3pi/24) + (pi/24) = 4pi/24 = pi/6
So, the equation becomes:
sin(pi/6) = sin(pi/8)cos(pi/24) + cos(pi/8)sin(pi/24)
Step 4: Evaluate the trigonometric functions for pi/6
sin(pi/6) is equal to 1/2, according to the unit circle.
So, the exact value for (sin(pi/8))(cos(pi/24)) + (cos(pi/8))(sin(pi/24)) is 1/2.
To find the exact value of the expression (sin(pi/8))(cos(pi/24)) + (cos(pi/8))(sin(pi/24)), you can utilize the trigonometric identity for the sum of angles:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Let's rewrite the expression:
(sin(pi/8))(cos(pi/24)) + (cos(pi/8))(sin(pi/24))
= sin(pi/8)cos(pi/24) + cos(pi/8)sin(pi/24)
Comparing this to the equation sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can see that:
A = pi/8
B = pi/24
Now, we can substitute the values of A and B into the equation:
sin(pi/8 + pi/24) = sin(pi/8)cos(pi/24) + cos(pi/8)sin(pi/24)
We can simplify the expression inside the sine function:
pi/8 + pi/24 = (3pi + pi)/24 = 4pi/24
Notice that 4pi/24 can be further simplified:
4pi/24 = pi/6
Now, we have:
sin(pi/6) = sin(pi/8)cos(pi/24) + cos(pi/8)sin(pi/24)
The exact value of sin(pi/6) is 1/2. Hence, the final answer is:
1/2