cos 23pi/12
i tried doing it and i got 0.994.. is that right? can someone explain how to do this to me? thanks!
it's to find the exact value..sorry i didn't mentioned before and thanks
cos 23pi/12 = cos (2pi - pi/12) = cos pi/12
4th quadrant
cos pi/12 = cos [(pi/6)/2] = sqrt((1 + cos pi/6)/2)
= sqrt((1+√3/2)/2)
= 0.9659
You could have used any good calculator to see whether your answer was right. 23pi/12 = -15 degrees
To find the value of cos(23π/12), we can use the unit circle and the properties of cosine.
First, let's determine the reference angle. To do that, we need to convert 23π/12 to its equivalent value within the interval [0, 2π].
Since 2π is equivalent to a full circular revolution, we can subtract 2π from 23π/12 until we get a value within that interval:
23π/12 - 2π = 23π/12 - 24π/12 = -π/12
So, -π/12 is the equivalent angle within the interval [0, 2π].
Next, let's identify the quadrant where this angle falls using the unit circle. The unit circle is divided into four quadrants and can help us determine the sign of the trigonometric function.
Since -π/12 is negative but very close to zero, it falls in the fourth quadrant.
In the fourth quadrant:
- sine (sin) is positive
- cosine (cos) is negative
- tangent (tan) is positive
Now, let's find the cosine value of -π/12 using the reference angle.
Using the Pythagorean identity, sin²θ + cos²θ = 1, we can find the value of sin(-π/12) as follows:
sin²(-π/12) + cos²(-π/12) = 1
Let's substitute cos(-π/12) with x:
sin²(-π/12) + x² = 1
Since we know that sin(-π/12) is positive and cos(-π/12) is negative, we can write:
sin(-π/12) = sqrt(1 - cos²(-π/12))
Now, substitute the value of sin(-π/12) with sqrt(1 - cos²(-π/12)):
(sqrt(1 - cos²(-π/12)))² + cos²(-π/12) = 1
1 - cos²(-π/12) + cos²(-π/12) = 1
1 = 1
This confirms that the value we obtained for the cosine of -π/12 is correct.
Therefore, cos(-π/12) ≈ 0.994, which means your answer of 0.994 is indeed correct.