Hi i have two questions:
what is the exact value of cos 5(pie)/12?
what is the exact value of cos2(pie)/9 cos(pie)/18 + sin2(pie)/9 sin(pie)/18?
thank you
first of all, the name of the Greek letter is pi not pie.
cos 5π/12 = (√3-1) / 2√2
cos2π/9 cosπ/18 + sin2π/9 sinπ/18 = cos(2π/9 - π/18) = cos π/6 = √3/2
To find the exact value of trigonometric functions like cosine, we can use various mathematical identities and properties. Let's go step by step to solve each question:
Question 1: What is the exact value of cos 5(pi)/12?
To find the exact value of cos 5(pi)/12, we can use the double-angle formula for cosine, which states that cos(2x) = 2cos^2(x) - 1.
1. First, we need to express the angle 5(pi)/12 as a sum or difference of angles whose exact values we know.
5(pi)/12 = 2(pi)/12 + 3(pi)/12 = pi/6 + pi/4
2. Using the sum-to-product identity for cosine, which states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can rewrite our expression:
cos (5(pi)/12) = cos (pi/6 + pi/4)
3. Applying the double-angle formula for cosine:
cos (5(pi)/12) = 2cos^2((pi/6 + pi/4)/2) - 1
4. Now it's a matter of simplifying the expression:
cos (5(pi)/12) = 2cos^2((5/24(pi))) - 1
= 2cos^2(5(pi)/24) - 1
So, the exact value of cos 5(pi)/12 is 2cos^2(5(pi)/24) - 1.
Question 2: What is the exact value of cos^2(pi)/9 cos(pi)/18 + sin^2(pi)/9 sin(pi)/18?
To find the exact value of this expression, we can use the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1.
1. Start by replacing sin^2(pi)/9 with 1 - cos^2(pi)/9:
cos^2(pi)/9 cos(pi)/18 + (1 - cos^2(pi)/9) sin(pi)/18
2. Now, substitute cos^2(pi)/9 with (1/2 + 1/2cos(2(pi)/9)):
(1/2 + 1/2cos(2(pi)/9)) cos(pi)/18 + (1 - (1/2 + 1/2cos(2(pi)/9))) sin(pi)/18
3. Simplify the expression further:
(1/2cos(pi)/18 + 1/2cos(2(pi)/9)cos(pi)/18) + (1/18sin(pi)/18 - 1/2cos(2(pi)/9)sin(pi)/18)
4. Combining like terms:
1/2cos(pi)/18 + 1/36cos(2(pi)/9) + 1/18sin(pi)/18 - 1/18cos(2(pi)/9)sin(pi)/18
So, the exact value of cos^2(pi)/9 cos(pi)/18 + sin^2(pi)/9 sin(pi)/18 is 1/2cos(pi)/18 + 1/36cos(2(pi)/9) + 1/18sin(pi)/18 - 1/18cos(2(pi)/9)sin(pi)/18.
Remember, when dealing with trigonometric functions, it's important to be familiar with the various identities and formulas that can be applied for simplification and finding exact values.