Let P be the point on the unit circle U that corresponds to t. Find the coordinates of P and the exact values of the trigonometric functions of t, whenever possible. (If there is no solution, enter NO SOLUTION.)
4pi = ( , )
sin(4pi) =
cos(4pi) =
tan(pi) =
cot(pi) =
sec(4pi) =
csc(pi) =
To find the coordinates of point P on the unit circle that corresponds to t, we can use the relationship between the unit circle and the trigonometric functions.
For any angle θ in radians on the unit circle, the x-coordinate of the corresponding point is given by cos(θ), and the y-coordinate is given by sin(θ).
1. 4π = (cos(4π), sin(4π))
The x-coordinate is cos(4π), and the y-coordinate is sin(4π).
2. sin(4π) = ?
To find sin(4π), we substitute 4π into the sine function:
sin(4π) = sin(2π + 2π) = sin(2π) = 0
So, sin(4π) = 0.
3. cos(4π) = ?
To find cos(4π), we substitute 4π into the cosine function:
cos(4π) = cos(2π + 2π) = cos(2π) = 1
So, cos(4π) = 1.
4. tan(π) = ?
To find tan(π), we substitute π into the tangent function:
tan(π) = sin(π) / cos(π) = 0 / (-1) = 0
So, tan(π) = 0.
5. cot(π) = ?
To find cot(π), we substitute π into the cotangent function:
cot(π) = cos(π) / sin(π) = (-1) / 0 = NO SOLUTION
cot(π) is undefined because sin(π) is equal to 0, and division by 0 is not possible.
6. sec(4π) = ?
To find sec(4π), we substitute 4π into the secant function:
sec(4π) = 1 / cos(4π) = 1 / 1 = 1
So, sec(4π) = 1.
7. csc(π) = ?
To find csc(π), we substitute π into the cosecant function:
csc(π) = 1 / sin(π) = 1 / 0 = NO SOLUTION
csc(π) is undefined because sin(π) is equal to 0, and division by 0 is not possible.
So, the coordinates of point P on the unit circle corresponding to t = 4π are (1, 0), and the exact values of the trigonometric functions are as follows:
sin(4π) = 0
cos(4π) = 1
tan(π) = 0
cot(π) = NO SOLUTION
sec(4π) = 1
csc(π) = NO SOLUTION
To find the coordinates of point P on the unit circle that corresponds to t, we can use the parametric equations for the unit circle:
x = cos(t)
y = sin(t)
In this case, t = 4π.
Step 1: Finding the coordinates of P
Substituting t = 4π into the parametric equations for the unit circle:
x = cos(4π)
y = sin(4π)
Step 1: Finding the coordinates of P
Substituting t = 4π into the parametric equations for the unit circle:
x = cos(4π) = 1
y = sin(4π) = 0
The coordinates of P are (1, 0).
Now let's find the exact values of the trigonometric functions of t = 4π.
Step 2: Finding the exact values of the trigonometric functions of t = 4π
sin(4π):
Using the unit circle, we can see that sin(4π) = 0.
cos(4π):
Also using the unit circle, we can see that cos(4π) = 1.
tan(π):
Using the unit circle, we can see that tan(π) = 0.
cot(π):
Using the reciprocal definitions of trigonometric functions, cot(π) = 1/tan(π) = 1/0. Since division by 0 is undefined, there is no solution.
sec(4π):
sec(θ) = 1/cos(θ), so sec(4π) = 1/cos(4π) = 1/1 = 1.
csc(π):
Using the reciprocal definitions of trigonometric functions, csc(π) = 1/sin(π) = 1/0. Since division by 0 is undefined, there is no solution.
The answers are:
sin(4π) = 0
cos(4π) = 1
tan(π) = 0
cot(π) = NO SOLUTION
sec(4π) = 1
csc(π) = NO SOLUTION