Suppose that termial point by t is the point (3/4, (square root 7)/4) on the unit circle. Find the terminal point determined by each of the following.
a) pi - t
b) -t
c) pi + t
d) 2pi + t
a) same height, opposite x so (-3/4, sqrt(7)/4)
b) same x,, opposite y so (3/4,-sqrt(7)/4)
c) opposite x, opposite y
d) same as original, exactly once around the circle
a) Pi - t: Well, if Pi is the number of pies I've eaten, then Pi minus t would be the number of pies I have left. So, the terminal point determined by Pi - t is probably (no pies left, because I ate them all, sorry).
b) -t: It seems like t is having a bit of a negative day here. The terminal point determined by -t would be a reflection of t across the x-axis. So, it's like t got a bad haircut and is feeling a bit down, poor guy.
c) Pi + t: Ah, the good old Pi and t combo. The terminal point determined by Pi + t is like adding extra toppings to your pizza. It's like getting a bonus point on a test you thought you failed. It's like finding money in your pocket you forgot about. In other words, it's a happy surprise!
d) 2Pi + t: Wow, t is really going for the record here! The terminal point determined by 2Pi + t is like t taking a victory lap around the unit circle. It's like t saying, "I conquered the circle once, now let's do it again!" You go, t!
To find the terminal point determined by each of the given angles, we can use the unit circle.
The terminal point determined by an angle t on the unit circle is given by (cos(t), sin(t)).
Given: terminal point by t = (3/4, √7/4)
a) To find the terminal point determined by π - t, we substitute π - t into the coordinates of the terminal point.
Substituting, we get:
(cos(π - t), sin(π - t))
Using the angle subtraction formula, we have:
(cos(π)cos(t) + sin(π)sin(t), sin(π)cos(t) - cos(π)sin(t))
Since cos(π) = -1 and sin(π) = 0, the expression simplifies to:
(-cos(t), -sin(t))
Therefore, the terminal point determined by π - t is (-cos(t), -sin(t)).
b) To find the terminal point determined by -t, we substitute -t into the coordinates of the terminal point.
Substituting, we get:
(cos(-t), sin(-t))
Since cos(-t) = cos(t) and sin(-t) = -sin(t), the terminal point determined by -t is (cos(t), -sin(t)).
c) To find the terminal point determined by π + t, we substitute π + t into the coordinates of the terminal point.
Substituting, we get:
(cos(π + t), sin(π + t))
Using the angle addition formula, we have:
(cos(π)cos(t) - sin(π)sin(t), sin(π)cos(t) + cos(π)sin(t))
Since cos(π) = -1 and sin(π) = 0, the expression simplifies to:
(-cos(t), sin(t))
Therefore, the terminal point determined by π + t is (-cos(t), sin(t)).
d) To find the terminal point determined by 2π + t, we substitute 2π + t into the coordinates of the terminal point.
Substituting, we get:
(cos(2π + t), sin(2π + t))
Since cos(2π + t) = cos(t) and sin(2π + t) = sin(t), the terminal point determined by 2π + t is (cos(t), sin(t)).
To summarize:
a) Terminal point determined by π - t: (-cos(t), -sin(t))
b) Terminal point determined by -t: (cos(t), -sin(t))
c) Terminal point determined by π + t: (-cos(t), sin(t))
d) Terminal point determined by 2π + t: (cos(t), sin(t))
To find the terminal point determined by a given angle, we can use the unit circle trigonometric values and the coordinates of the given terminal point (3/4, √7/4).
First, let's recall some important information about the unit circle. The unit circle has a radius of 1 and is centered at the origin (0, 0). At any angle θ in standard position (i.e., starting from the positive x-axis and rotating counterclockwise), the x-coordinate of the terminal point on the unit circle is given by cos(θ), and the y-coordinate is given by sin(θ).
Now, let's analyze each case:
a) To find the terminal point determined by π - t:
Substitute π - t into the trigonometric functions for x and y coordinates:
x = cos(π - t) = -cos(t)
y = sin(π - t) = sin(t)
Since we are given the x and y coordinates of t (3/4, √7/4), we can find the new coordinates for x and y by substituting t into the expressions:
x = -cos(t)
y = sin(t)
b) To find the terminal point determined by -t:
Substitute -t into the trigonometric functions for x and y coordinates:
x = cos(-t) = cos(t)
y = sin(-t) = -sin(t)
Using the given coordinates of t (3/4, √7/4), we can substitute t into the expressions to find the new coordinates for x and y:
x = cos(t)
y = -sin(t)
c) To find the terminal point determined by π + t:
Substitute π + t into the trigonometric functions for x and y coordinates:
x = cos(π + t) = -cos(t)
y = sin(π + t) = -sin(t)
Using the given coordinates of t (3/4, √7/4), we can substitute t into the expressions to find the new coordinates for x and y:
x = -cos(t)
y = -sin(t)
d) To find the terminal point determined by 2π + t:
Substitute 2π + t into the trigonometric functions for x and y coordinates:
x = cos(2π + t) = cos(t)
y = sin(2π + t) = sin(t)
Using the given coordinates of t (3/4, √7/4), we can substitute t into the expressions to find the new coordinates for x and y:
x = cos(t)
y = sin(t)
In summary:
a) The terminal point determined by π - t is (-cos(t), sin(t)).
b) The terminal point determined by -t is (cos(t), -sin(t)).
c) The terminal point determined by π + t is (-cos(t), -sin(t)).
d) The terminal point determined by 2π + t is (cos(t), sin(t)).
Note that the values of cos(t) and sin(t) can be obtained from the coordinates of t, which are (3/4, √7/4), by considering the ratios of the sides of the right triangle formed by dropping a perpendicular from the terminal point on the unit circle to the x-axis.