How do you figure out the exact circular function value of sec(23pie/6)?
*I don't know how to get the [pie]symbol on my keyboard
23pi/6 is pi/6 short of two rotations of 4pi
so 23pi/6 is coterminal with -pi/6
(23pi/6 = 690º. 690º - 360º = 330 or -30º)
sec (23pi/6) = sec (-pi/6)
= 1/cos(-pi/6)
= 1/cos(pi/6)
= 1/(√3/2)
= 2/√3 or 2√3/2 after rationalizing the denominator
To find the exact circular function value of sec(23π/6), we can start by using the unit circle.
1. First, let's determine the angle in degrees. Multiply the fraction by 180/π (since there are 180 degrees in π radians):
23π/6 * (180/π) = (23 * 180)/6 = 23 * 30 = 690 degrees
2. Next, let's convert the angle to its equivalent in the first revolution (360 degrees) by subtracting multiples of 360:
690 degrees - 1 * 360 degrees = 690 - 360 = 330 degrees
3. Since secant is the reciprocal of cosine, we need to find the value of cosine for the angle 330 degrees. From the unit circle, we know that cosine is positive in the fourth quadrant (180-270 degrees range).
4. To find the cosine value, we can use the 30-60-90 triangle within the fourth quadrant. Since it's a 30-60-90 triangle, the cosine value is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side is √3/2, and the hypotenuse is 1:
cos(330 degrees) = √3/2 / 1 = √3/2
5. Finally, since secant is the reciprocal of cosine, we can find the secant value:
sec(330 degrees) = 1 / cos(330 degrees) = 1 / (√3/2) = 2 / √3
Therefore, the exact circular function value of sec(23π/6) is 2 / √3.
To find the exact circular function value of sec(23π/6), we need to use the unit circle and some trigonometric identities.
First, let's address the symbol you mentioned. The symbol you're referring to is "π," which is the Greek letter Pi used to represent the mathematical constant approximately equal to 3.14159. If you don't have the Pi symbol on your keyboard, you can simply type "pi" or "PI" to represent it.
Now, let's proceed with finding the value of sec(23π/6):
1. Start by drawing a unit circle. A unit circle has a radius of 1 and is centered at the origin (0,0) on a coordinate plane.
2. The angle 23π/6 is in radians. Remember that the circumference of a circle is 2π, so 23π/6 corresponds to approximately 3.83 full revolutions counterclockwise around the unit circle.
3. To find the exact value of sec(23π/6), we need to determine the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
4. Since 23π/6 represents more than 2 full revolutions, you can subtract 2π (or 12π/6) from it to get an equivalent angle within the first two revolutions. Thus, we have: 23π/6 - 12π/6 = 11π/6.
5. The angle 11π/6 corresponds to the same position on the unit circle as the angle π/6 (30 degrees) since their terminal sides are at the same location.
6. The x-coordinate of the point where the terminal side of π/6 intersects the unit circle is √3/2.
7. Since sec(x) is the reciprocal of cos(x), we can find the value of sec(11π/6) by taking the reciprocal of cos(π/6): sec(11π/6) = 1/cos(π/6).
8. The cosine of π/6 is √3/2. Substituting this value, we have: sec(11π/6) = 1/(√3/2).
9. To divide by a fraction, you multiply by its reciprocal. Therefore, we can rewrite the expression as: sec(11π/6) = 1 * (2/√3).
10. Simplifying further, we get: sec(11π/6) = 2/√3.
11. To rationalize the denominator, we multiply the numerator and denominator by √3: sec(11π/6) = (2/√3) * (√3/√3) = 2√3/3.
Therefore, the exact circular function value of sec(23π/6) is 2√3/3.