find the exact value of cos 3.14/8
Since 3.14 is an approximation to π,
I suppose you mean finding the exact value of cos(π/8).
Use the half angle formulae:
cos²(x/2)=(1+cos(x))/2...(1)
sin²(x/2)=(1-cos(x))/2...(2)
The second formula is not required for this problem, but I suggest you memorize both of them, because they will be very useful in your academic life.
Start from known values,
cos(π/2)=0
cos²(π/4)=(1+0)/2
cos(π/4)=(√2)/2
Apply the formula one more time to get cos(π/8):
cos²(π/8)=(1+(√2)/2))/2
cos(π/8)=√(2+√2)/2
Okay, I understand how to get pi/8 but how do you figure pi/7 and how do you figure cos(10degrees) when you can use the sum and difference method of angles? Please help it is driving me crazy, and I have finals coming up. Need to get this down. Can I do the method about using pi/3 and pi/4 in the half angle formulas to find exact value of cos and sin of (pi/7)? I am so confused.
To find the exact value of cos(3.14/8), we can use the half-angle formula for cosine.
The half-angle formula for cosine is given by:
cos(x/2) = ± sqrt((1 + cos(x))/2)
By substituting x = 3.14/4, we can calculate cos(3.14/8) as follows:
cos(3.14/8) = ± sqrt((1 + cos(3.14/4))/2)
Now, we need to calculate cos(3.14/4):
cos(3.14/4) = sqrt(2)/2
Substituting this value back into the half-angle formula, we get:
cos(3.14/8) = ± sqrt((1 + sqrt(2)/2)/2)
Simplifying further:
cos(3.14/8) = ± sqrt((2 + sqrt(2))/4)
The final exact value of cos(3.14/8) is ± sqrt((2 + sqrt(2))/4), which means there are two possible solutions: positive and negative, depending on the context or problem you are working on.