if sine theta=.3416 and theta is in quadrant I what is cos theta/2? Using double angle identities.
cos 2θ = 2cos^2θ - 1, so
cos θ/2 = √((1+cosθ)/2)
since sinθ = .3416, cosθ = .9398
so, cos θ/2 = √((1+.9398)/2) = .9848
check: θ = 19.97°
cos 9.985° = .9848
To find the value of cos(theta/2) using the double angle identities, we can utilize the identity:
cos(2x) = 2cos^2(x) - 1
In this case, we are given the value of sin(theta) = 0.3416. Because theta is in Quadrant I, both sin(theta) and cos(theta) are positive.
Step 1: Find cos(theta) using the Pythagorean identity.
Since sin(theta) = 0.3416, we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Plugging in the value of sin(theta) = 0.3416:
(0.3416)^2 + cos^2(theta) = 1
0.1166 + cos^2(theta) = 1
cos^2(theta) = 1 - 0.1166
cos^2(theta) = 0.8834
cos(theta) = √(0.8834)
cos(theta) ≈ 0.9387
Step 2: Use the double angle identity to find cos(theta/2).
Let's substitute x = theta/2 into the double angle formula:
cos(2x) = 2cos^2(x) - 1
cos(theta) = 2cos^2(theta/2) - 1
Now, substitute the value of cos(theta) we found earlier:
0.9387 = 2cos^2(theta/2) - 1
Rearrange the equation:
2cos^2(theta/2) = 0.9387 + 1
2cos^2(theta/2) = 1.9387
Divide both sides by 2:
cos^2(theta/2) = 0.96935
cos(theta/2) = √(0.96935)
cos(theta/2) ≈ 0.9846
Therefore, cos(theta/2) ≈ 0.9846 using the double angle identities.