1. Find the exact value of the expression.
cos pi/16 * cos 3pi/16 - sin pi/16 * sin 3pi/16
My ans: sqrt2/2
2. Find the exact value of the sin of the angle.
17pi/12 = 7pi/6 + pi/4
My ans: (sqrt2 - sqrt6)/4
Hmmm. I get
-(√2+√6)/4
Better check your +- signs.
To find the exact value of the expression cos(pi/16) * cos(3pi/16) - sin(pi/16) * sin(3pi/16), we can use the trigonometric identity for the cosine of the difference of angles:
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
In this case, A = pi/16 and B = 3pi/16. Plugging these values into the identity, we have:
cos(pi/16 - 3pi/16) = cos(pi/16) * cos(3pi/16) + sin(pi/16) * sin(3pi/16)
The left side simplifies to:
cos(-pi/4) = cos(pi/4)
Since cosine is an even function, cos(-pi/4) is equal to cos(pi/4). So we have:
cos(pi/4) = cos(pi/16) * cos(3pi/16) + sin(pi/16) * sin(3pi/16)
Using the values of cos(pi/4) and rearranging the equation, we get:
sqrt(2)/2 = cos(pi/16) * cos(3pi/16) + sin(pi/16) * sin(3pi/16)
Therefore, the exact value of the expression is sqrt(2)/2.
For the second question, we need to find the exact value of sin(17pi/12) using the given information 17pi/12 = 7pi/6 + pi/4.
We can use the addition formula for sine to solve this:
sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B)
In this case, A = 7pi/6 and B = pi/4. Plugging these values into the identity, we have:
sin(7pi/6 + pi/4) = sin(7pi/6) * cos(pi/4) + cos(7pi/6) * sin(pi/4)
Since we know the values of sin(7pi/6) = -sqrt(3)/2 and cos(pi/4) = sqrt(2)/2, we can substitute them in:
sin(7pi/6 + pi/4) = (-sqrt(3)/2) * (sqrt(2)/2) + cos(7pi/6) * sin(pi/4)
Similarly, we know that cos(7pi/6) = -sqrt(3)/2 and sin(pi/4) = sqrt(2)/2, so we can substitute those values as well:
sin(7pi/6 + pi/4) = (-sqrt(3)/2) * (sqrt(2)/2) + (-sqrt(3)/2) * (sqrt(2)/2)
Simplifying this expression, we get:
sin(7pi/6 + pi/4) = -(sqrt(6) + sqrt(2)) / 4
Therefore, the exact value of sin(17pi/12) is -(sqrt(6) + sqrt(2)) / 4.