For the following values of t find (a) the reference number of t and (B) the terminal point determined by t
7pie/3
To find the reference number of t, we need to find an angle that is coterminal with 7π/3 and lies between 0 and 2π.
Since 2π is the entire circle, we can subtract 2π from 7π/3 until we obtain an angle between 0 and 2π.
7π/3 - 2π = π/3
Now, we have an angle of π/3.
To find the terminal point determined by t, we use the unit circle.
Since π/3 is in the first quadrant, we move counterclockwise from the positive x-axis by an angle of π/3.
This gives us the terminal point (cos(π/3), sin(π/3)) = (1/2, √3/2).
Therefore, for t = 7π/3, the reference number is π/3 and the terminal point is (1/2, √3/2).
To find the reference number and terminal point determined by t = 7π/3, we can use the following steps:
Step 1: Convert t to radians
t = 7π/3
Step 2: Simplify the fraction (if possible)
t = (7π)/(3) = (7/3)π
Step 3: Determine the reference number
To find the reference number, we need to convert t to an angle between 0 and 2π (or between 0 and 360 degrees). We can do this by using the following formula:
Reference number = t modulus 2π
Since t = (7/3)π, we have:
Reference number = (7/3)π % (2π)
Simplifying the expression:
Reference number = (7π/3) % (2π)
The modulus operator "%" gives us the remainder when (7π/3) is divided by (2π). In this case, the remainder is (π/3), so:
Reference number = π/3
Step 4: Determine the terminal point
The terminal point is determined by the reference number on the unit circle. Since the reference number is π/3, we can find the terminal point by converting the angle to Cartesian coordinates using sine and cosine.
The coordinates (x, y) for the terminal point on the unit circle are:
x = cos(Reference number)
y = sin(Reference number)
For π/3, we have:
x = cos(π/3) ≈ 0.5
y = sin(π/3) ≈ √3/2
So the terminal point determined by t = 7π/3 is approximately (0.5, √3/2).
To summarize:
(a) The reference number of t = 7π/3 is π/3.
(b) The terminal point determined by t = 7π/3 is approximately (0.5, √3/2).
To find the reference number (a) and the terminal point (B) determined by t = 7π/3, let's break down the process step by step.
Step 1: Calculating the Reference Number (a)
The reference number of an angle is the equivalent angle within one revolution (360 degrees) that has the same terminal side. To calculate the reference number (a), we need to find the angle within one revolution that is equivalent to t.
- Since 1 revolution equals 2π radians, we will divide t by 2π to get the number of complete rotations.
- Divide 7π/3 by 2π:
(7π/3) / (2π) = 7/6
The reference number (a) of t = 7π/3 is 7/6, indicating that the terminal point of t lies on the 7/6 revolution.
Step 2: Finding the Terminal Point (B)
To find the terminal point determined by t = 7π/3, we need to plot it on the unit circle.
- Start with the initial side of the angle, which is the positive x-axis (the rightmost point on the unit circle).
- Count counter-clockwise (since the angle is positive) by a number of radians equal to the absolute value of t.
Since t = 7π/3, the terminal point is 7π/3 radians (counterclockwise) from the positive x-axis.
For visual reference, consider the unit circle as follows:
π/6
π/3 | π/4
π/2 | π/2
________|_______
| t = (7π/3)
|
3π/2
|
|π
Counting (7π/3) radians counterclockwise from the positive x-axis (0 radians), we reach the terminal point, which corresponds to the t value.
Thus, the terminal point determined by t = 7π/3 is located at an angle of (7π/3) radians counterclockwise from the positive x-axis on the unit circle.