Use a half-angle formula to find the exact value of cos 5ð/12
your symbol for pi did not come out, just type pi, that seems to be the simplest way.
A lot of students seem to find it easier to think in degrees,
5pi/12 = 75º
cos(5pi/12) radians
= cos 75º
= cos(45 + 30)
= cos45cos30 - sin45sin30
= (√2/2)(√3/2) - √2/2)(1/2)
= (√6 - √2)/4
To find the exact value of cos 5π/12 using a half-angle formula, we can start by using the formula for cos(2θ):
cos(2θ) = 2cos²(θ) - 1
Let's rewrite the angle 5π/12 as a double angle:
5π/12 = 2(5π/24)
Now, we can use the half-angle formula for cos(θ):
cos(θ/2) = ±√((1 + cos(θ))/2)
In this case, θ = 5π/24, so let's substitute it in:
cos(5π/24) = ±√((1 + cos(5π/12))/2)
Now, we need to solve for cos(5π/12) using the half-angle formula. To do this, we need to use another trigonometric identity called the double-angle formula for cos(2θ):
cos(2θ) = cos²(θ) - sin²(θ)
Using this formula, we can get cos(5π/12) in terms of cos(5π/24):
cos(5π/12) = cos²(5π/24) - sin²(5π/24)
Since we have cos²(5π/24) in the half-angle formula, we can solve for it in terms of cos(5π/12):
cos²(5π/24) = (1 + cos(5π/12))/2
Now, let's substitute this back into the half-angle formula:
cos(5π/24) = ±√((1 + (1 + cos(5π/12))/2)/2)
Simplifying this expression, we get:
cos(5π/24) = ±√((3 + cos(5π/12))/4)
Finally, we can substitute this value back into the half-angle formula for cos(θ/2):
cos(5π/12) = ±√((1 + cos(5π/12))/2) × √((3 + cos(5π/12))/4)
Simplifying this further will give us the exact value of cos 5π/12.