Please help me solve this problem. I have a test tomorrow and I do not understand at all.
Determine all solutions of the equations in radians.
Find cos x/2, given that cos x = 1/4 and x terminates in 0 < x < π/2.
let's use
cos 2A = 2cos^2 A - 1
or
cos x = 2cos^2(x/2) - 1
1/4 = 2cos^2(x/2) - 1
5/8 = cos^2(x/2)
cosx = √5/√8 or √10/4
How did you get 5/8? Thanks for your help!
from
1/4 = 2cos^2(x/2) - 1
1/ 4 + 1 = 2cos^2(x/2)
5/4 = 2cos^2(x/2) , now divide by 2
5/8 = cos^2(x/2) , now take √
cos x/2 = √5/√8 , rationalizing
cos x/2 = (√5/√8)(√8/√8) = √40/8 = 2√10/8 = √10/4
To find the value of cos(x/2), given that cos(x) = 1/4 and x terminates in 0 < x < π/2, we can use the double-angle formula for cosine:
cos(2θ) = 2cos^2(θ) - 1
The double-angle formula allows us to find the value of cos(x/2) using the given value of cos(x). Here's how you can do it step-by-step:
Step 1: Given the value of cos(x) = 1/4, we need to find the value of cos(x/2).
Step 2: Use the double-angle formula to find cos(2x):
cos(2x) = 2cos^2(x) - 1.
Step 3: Substitute the value of cos(x) from the given equation into the double-angle formula:
cos(2x) = 2(1/4)^2 - 1.
Step 4: Simplify the equation:
cos(2x) = 1/8 - 1 = -7/8.
Step 5: Use the half-angle formula to find cos(x/2):
cos(x/2) = ±√((1 + cos(2x))/2).
Step 6: Substitute the value of cos(2x) calculated in Step 4 into the half-angle formula:
cos(x/2) = ±√((1 + (-7/8))/2).
Step 7: Simplify the equation:
cos(x/2) = ±√((1 - 7/8)/2) = ±√(1/8) = ±√(1/(2^3)) = ±(1/√8) = ±(1/2√2).
Step 8: Further simplify the equation:
cos(x/2) = ±(1/2√2) = ±(1/(2√2)) = ±(1/(2√(2*2))) = ±(1/(2*2)).
Step 9: Simplify the equation:
cos(x/2) = ±(1/4).
Therefore, the value of cos(x/2) is ±(1/4).