For the following values of t find (a) the reference number of t and (B) the terminal point determined by t
-11 pie/6
To find the reference number of -11pi/6, we need to find the equivalent angle within one revolution (2pi) that has the same terminal side.
To do this, we can add or subtract multiples of 2pi until we get an angle between 0 and 2pi.
In this case, we can add 2pi to -11pi/6 repeatedly until we get a value within the range of 0 to 2pi.
(-11pi/6) + (12pi/6) = pi/6
Thus, the reference number of -11pi/6 is pi/6.
To find the terminal point determined by -11pi/6, we can use the unit circle.
Starting from the positive x-axis (the reference angle of 0), we can count counter-clockwise by adding the angle -11pi/6 to the reference angle.
pi/6 + (-11pi/6) = -10pi/6
The terminal point determined by -11pi/6 is on the negative x-axis, specifically (-1, 0).
To find the reference number and terminal point for the value of t = -11π/6, we can follow these steps:
Step 1: Determine the reference angle.
The reference angle is the positive angle measured between the terminal side of an angle and the x-axis in standard position. We can find the reference angle by taking the absolute value of the given angle.
Given angle: t = -11π/6
Absolute value of -11π/6: |-11π/6| = 11π/6
So, the reference angle is 11π/6.
Step 2: Determine the reference number.
The reference number is the equivalent angle in the range [0, 2π]. To find the reference number, we can find the equivalent angle for the reference angle.
Equivalent angle for 11π/6:
11π/6 is greater than 2π, so we need to subtract 2π from it until we get an angle in the range [0, 2π].
11π/6 - 2π = 11π/6 - 12π/6 = -π/6
Therefore, the reference number for -11π/6 is -π/6.
Step 3: Determine the terminal point.
The terminal point is the point on the unit circle corresponding to the given angle.
To find the terminal point, we can use the coordinate representations of the unit circle.
Unit Circle:
The unit circle is a circle with a radius of 1, centered at the origin (0, 0) in the Cartesian plane.
Coordinates on Unit Circle:
At -π/6 or 330°, the x-coordinate is cos(-π/6) = √3/2
At -π/6 or 330°, the y-coordinate is sin(-π/6) = -1/2
So, the terminal point determined by t = -11π/6 is (√3/2, -1/2).
In summary:
(a) The reference number of t = -11π/6 is -π/6.
(b) The terminal point determined by t = -11π/6 is (√3/2, -1/2).
To find the reference number of t, we need to determine the equivalent angle in the interval [0, 2π). The formula to find the reference number is:
reference number = t mod (2π)
In this case, t = -11π/6, so applying the formula:
reference number = (-11π/6) mod (2π)
To simplify this computation, let's convert π into degrees. Since 180° = π radians, we can replace π with 180° in the equation:
reference number = (-11 * 180°/6) mod (2 * 180°)
reference number = (-11 * 30°) mod (360°)
Now, let's calculate the reference number:
reference number = -330° mod 360°
To find the terminal point determined by t, we need to locate the angle t on the unit circle and determine its coordinates (x, y). The formula to find the terminal point using radians is:
terminal point = (cos(t), sin(t))
Let's calculate the terminal point:
terminal point = (cos(-11π/6), sin(-11π/6))
First, let's convert -11π/6 into degrees:
-11π/6 = (-11 * 180°) / 6 = -330°
Then, we determine the coordinates:
terminal point = (cos(-330°), sin(-330°))
Now, we can either use a calculator to find the cosine and sine of -330° or use trigonometric identities to simplify the calculation (for instance, using reference angles).
By using trigonometric identities, we know that cos(-330°) = cos(30°) and sin(-330°) = -sin(30°) because cosine is an even function (same value in negative angles) while sine is an odd function (negative sign in negative angles).
Thus, we can calculate the terminal point as:
terminal point = (cos(30°), -sin(30°))
Using the unit circle, we know that cos(30°) = √3/2 and sin(30°) = 1/2. Therefore, the terminal point is:
terminal point = (√3/2, -1/2)
To summarize:
(a) The reference number of t is -330°.
(b) The terminal point determined by t is (√3/2, -1/2).