What is the exact value of 11pi/8? (with steps please) :)

Not sure if you are better thinking in terms of degrees or radians, most people are more familiar with degrees

11π/8 radians = 247.5°

so cos(11π/8) = cos(247.5°) (in quad III)
= -cos 67.5

let that sit.

67.5 = 90-22.5
cos 67.5 = cos(90-22.5)
= sin 22.5 (complementary angles property, e.g. sin 20 = cos70 )

using cos 2A = 1 - 2sin^2 A
cos 45 = 1 - 2sin^2 (22.5)
√2 /2 = 1 - 2sin^2 (22.5)
2 sin^2 22.5 = 1 - √2/2 = (2-√2)2
sin^2 22.5 = (2-√2)/4
sin 22.5 = √(2-√2)/2

so recall we needed
- cos67.5
= -sin22.5
= - √(2-√2)/2

cos (11π/8) = -√(2-√2)/2

check:
-√(2-√2)/2 = appr. -.38268...
cos 247.5 = -.3826..
How about that ???

11π/8 is an exact value.

where does the trig part of your question come in?
Was is something like sin (11π/8) ?

yea what reiny said

Oh I'm sorry. It's cosine (11pi/8) :)

It's good but there is a typo... The line with the typo that does not make sense is:

2 sin^2 22.5 = 1 - √2/2 = (2-√2)2

...it should read:

2 sin^2 22.5 = 1 - √2/2 = (2-√2)/2

The whole process after that is legit.

(And here is the edited version, below:)

using cos 2A = 1 - 2sin^2 A
cos 45 = 1 - 2sin^2 (22.5)
√2 /2 = 1 - 2sin^2 (22.5)
2 sin^2 22.5 = 1 - √2/2 = (2-√2)/2
sin^2 22.5 = (2-√2)/4
sin 22.5 = √(2-√2)/2

so recall we needed
- cos67.5
= -sin22.5
= - √(2-√2)/2

cos (11π/8) = -√(2-√2)/2

To find the exact value of \( \frac{{11\pi}}{8} \), we can use the unit circle and trigonometric functions.

1. Start by drawing a unit circle, which is a circle with a radius of 1 unit.
2. Divide the circle into eight equal parts, representing the eight angles formed by \( \pi \) radians.
3. The angle \( \frac{{11\pi}}{8} \) is slightly larger than \( \pi \) radians, which means it falls between \( \pi \) and \( \frac{{3\pi}}{2} \) radians.
4. In the fourth quadrant (lower right quadrant), locate the reference angle that is formed by the terminal side of \( \frac{{11\pi}}{8} \) and the x-axis.
5. Calculate the reference angle by subtracting \( \pi \) from \( \frac{{11\pi}}{8} \).
- \( \frac{{11\pi}}{8} - \pi = \frac{{3\pi}}{8} \)
- Therefore, the reference angle is \( \frac{{3\pi}}{8} \).
6. Determine the trigonometric values for the reference angle.
- Sine (\( \sin \)): To find the sine value, look at the y-coordinate of the point where the terminal side intersects the circle. In this case, it is \( \sin{\left(\frac{{3\pi}}{8}\right)} \).
- Cosine (\( \cos \)): To find the cosine value, look at the x-coordinate of the point where the terminal side intersects the circle. In this case, it is \( \cos{\left(\frac{{3\pi}}{8}\right)} \).
- Tangent (\( \tan \)): To find the tangent value, divide the sine value by the cosine value. In this case, it is \( \tan{\left(\frac{{3\pi}}{8}\right)} = \frac{{\sin{\left(\frac{{3\pi}}{8}\right)}}}{{\cos{\left(\frac{{3\pi}}{8}\right)}}} \).
7. Finally, remember that the exact value of \( \frac{{11\pi}}{8} \) is equal to the negative of the trigonometric values of the reference angle because it is situated in the fourth quadrant.
- \( \sin{\left(\frac{{11\pi}}{8}\right)} = -\sin{\left(\frac{{3\pi}}{8}\right)} \)
- \( \cos{\left(\frac{{11\pi}}{8}\right)} = -\cos{\left(\frac{{3\pi}}{8}\right)} \)
- \( \tan{\left(\frac{{11\pi}}{8}\right)} = -\tan{\left(\frac{{3\pi}}{8}\right)} \)

Using these steps, you can obtain the exact value of \( \frac{{11\pi}}{8} \) by finding the trigonometric values of its reference angle, and then negating them to account for the position in the fourth quadrant.