For the following values of t find (a) the reference number of t and (B) the terminal point determined by t
17pie/4
To find the reference number of t, we need to find the angle that has the same initial side as t and intersects the positive x-axis.
Since one full revolution is equal to 2π radians, we can subtract 2π from t until we get an angle between 0 and 2π.
17π/4 = (16π + π/4)
Since 16π is equivalent to 8 full revolutions (2π * 8 = 16π), we can ignore that part.
So, π/4 is the reference number of t.
To find the terminal point determined by t, we use the unit circle.
For π/4, the cosine and sine of the angle can be found using the 45-45-90 triangle formed in the first quadrant of the unit circle.
Since the 45-45-90 triangle has sides in the ratio of 1:1:√2, the cosine and sine values for π/4 are both 1/√2.
Therefore, the terminal point determined by t is (1/√2, 1/√2).
To find the reference number and terminal point determined by t = 17π/4, we can follow these steps:
Step 1: Convert t to its reference angle
The reference angle is the angle between the terminal side of t and the x-axis.
To find the reference angle:
Reference angle = t - (2π) * floor(t / (2π))
where "floor" is the greatest integer function.
In this case,
Reference angle = (17π/4) - (2π) * floor((17π/4) / (2π))
Simplifying further:
Reference angle = (17π/4) - (2π) * floor(17/8)
Reference angle = (17π/4) - (2π) * 2
Reference angle = (17π/4) - (4π)
Reference angle = π/4
Step 2: Find the reference number
The reference number is the angle equivalent to the reference angle but within the range [0, 2π).
In this case,
Reference number = reference angle
Reference number = π/4
Step 3: Find the terminal point
The terminal point is determined by using the reference number on the unit circle. The terminal point will have coordinates (x, y) on the unit circle, where x and y represent the x-coordinate and y-coordinate, respectively.
Since the reference number is π/4 (45 degrees), the corresponding terminal point can be determined as:
x = cos(π/4)
y = sin(π/4)
Evaluating these trigonometric functions:
x = √2 / 2
y = √2 / 2
Therefore, the terminal point determined by t = 17π/4 is:
Terminal point = (x, y) = ( √2 / 2, √2 / 2 )
To summarize:
(a) The reference number of t = 17π/4 is π/4.
(b) The terminal point determined by t = 17π/4 is ( √2 / 2, √2 / 2 ).
To find the reference number of t, we can convert the given angle to standard position, which means rotating counterclockwise until the angle is between 0 and 2π (0 and 360°).
To convert 17π/4 to standard position:
1. Divide the given angle by 2π: (17π/4)/(2π) = 17/8.
2. The decimal value 17/8 is equivalent to 2.125 in decimal form. Since the angle is greater than 2, we need to subtract 2 to bring it between 0 and 2.
3. 2.125 - 2 = 0.125.
4. To convert the decimal value back to fraction form: 0.125 = 1/8.
Therefore, the reference number of 17π/4 is 1/8.
To find the terminal point determined by t, we can use the unit circle.
The unit circle is a circle of radius 1 centered at the origin (0, 0) on a coordinate plane. Each point on the unit circle corresponds to an angle measure.
First, recall that π (pi) represents half a circle (180°) and 2π represents a full circle (360°). Since 17π/4 is greater than 2π, we know it represents more than one full rotation.
To find the terminal point determined by t:
1. Determine the angle on the unit circle that corresponds to the reference number found earlier, which is 1/8. Since 1/8 is small, the terminal point will be close to the positive x-axis.
2. On the unit circle, locate the angle that corresponds to 1/8. This will be approximately π/4 (45°). The terminal point will be where the angle intersects the unit circle.
3. Since 17π/4 is equivalent to 4 and 1/4 of a full rotation counterclockwise, the terminal point will be 4 and 1/4 times the distance between the positive x-axis and the point of intersection.
4. Determine the coordinates of the terminal point by using the unit circle or trigonometry to calculate the x-coordinate and y-coordinate.
Therefore, to find the terminal point determined by t, more information is needed, such as whether you want the coordinates in Cartesian form or polar form, or if the angle is in radians or degrees.