If cosx = 10/19 an pi < x < 2pi, find the exact value of cos x/2

Use the double angle formula.

Cos 2Y= sin^2 Y - cos^2Y = 1-2Cos^2 Y.

let y= x/2, and 2Y=x

Why did the angle go to therapy? Because it had trouble adapting to the double angle formula!

Okay, let's dive into the problem. We have cos(x) = 10/19, and we want to find cos(x/2).

First, let's find cos(2Y) using the double angle formula:
cos(2Y) = 1 - 2cos^2(Y)

But wait, we need to express Y in terms of x/2! So, let's rewrite 2Y = x:
2(x/2) = x
Y = x/2

Now we can substitute Y back into the formula:
cos(2(x/2)) = 1 - 2cos^2(x/2)

But remember, we also have cos(x) = 10/19. So let's substitute cos^2(x/2) with cos^2(x)/4:
cos(2(x/2)) = 1 - 2(cos^2(x)/4)

Now, we need to find cos(x). Since we only know cos(x), not x directly, we can use the Pythagorean identity to find sin(x):
sin(x) = √(1 - cos^2(x))

Since we're specifically looking for cos(x/2), we'll need sin(x/2), which we can find using the half-angle formula:
sin(x/2) = ±√((1 - cos(x))/2)

Woohoo! We're getting closer to the answer. Now let's substitute sin(x) and sin(x/2) back into our equation:
cos(2(x/2)) = 1 - 2((1 - cos^2(x))/4)

Simplifying further:
cos(2(x/2)) = 1 - ((1 - cos^2(x))/2)

But remember, we still have cos(x) = 10/19. Let's put it all together:
cos(2(x/2)) = 1 - ((1 - (10/19)^2)/2)

Now, let's simplify:
cos(2(x/2)) = 1 - ((1 - 100/361)/2)

Still with me? Great! Simplifying further:
cos(2(x/2)) = 1 - ((361 - 100)/361)/2

Brace yourself, here comes the final answer:
cos(2(x/2)) = (361 - 100)/361/2

Phew! We made it! The exact value of cos(x/2) is (361 - 100)/361 divided by 2.

I hope I didn't twist your brain too much with all those angles and formulas. Keep smiling, just like a well-rounded clown!

To find the exact value of cos(x/2), we will use the double angle formula and substitute y = x/2 and 2Y = x.

The double angle formula for cos is:
cos(2Y) = 1 - 2sin^2(Y)

We are given that cos(x) = 10/19 and that pi < x < 2pi. This tells us that x lies in the second quadrant, where cosine is negative.

In the second quadrant, sin(Y) is positive. We can find sin(Y) using the identity sin^2(Y) + cos^2(Y) = 1:
sin^2(Y) = 1 - cos^2(Y)
sin^2(Y) = 1 - (10/19)^2
sin^2(Y) = 1 - 100/361
sin^2(Y) = 261/361
sin(Y) = √(261/361)
sin(Y) = √261/19

Now, let's substitute these values into the double angle formula and solve for cos(y):

cos(2Y) = 1 - 2sin^2(Y)
cos(x) = 1 - 2(sin(Y))^2
10/19 = 1 - 2(√261/19)^2
10/19 = 1 - 2(261/361)
10/19 = 1 - 522/361
10/19 = (361 - 522)/361
10/19 = -161/361

Since we know that cos(y) is negative in the second quadrant, we can conclude that cos(y) = -161/361.

Finally, let's find the value of cos(x/2) using the substitution y = x/2:

cos(x/2) = cos(y)
cos(x/2) = -161/361

Therefore, the exact value of cos(x/2) is -161/361.

To find the exact value of cos(x/2), we can use the double angle formula for cosine.

Let's start by finding the value of cos(2Y), where Y = x/2 and 2Y = x.

According to the double angle formula for cosine:
cos(2Y) = 1 - 2(cos^2Y)

Since we know that cos(x) = 10/19, we can substitute Y = x/2 into the equation:
cos(2(x/2)) = 1 - 2(cos^2(x/2))

Now, let's simplify the equation:
cos(x) = 1 - 2(cos^2(x/2))
cos(x) = 1 - 2(cos^2Y)

Since cos(x) = 10/19, we have:
10/19 = 1 - 2(cos^2Y)

Let's solve for cos^2Y:
cos^2Y = (1 - 10/19) / 2

Simplifying further:
cos^2Y = (19 - 10)/19 / 2
cos^2Y = 9/19 / 2
cos^2Y = (9/19) * (1/2)
cos^2Y = 9/38

Now, let's find the square root of cos^2Y to get the value of cosY:
cosY = sqrt(9/38)

Finally, we can find the value of cos(x/2) by substituting cosY back into the equation:
cos(x/2) = cosY = sqrt(9/38)

Therefore, the exact value of cos(x/2) is sqrt(9/38).