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.)
4ð = ( , )
sin(4ð) =
cos(4ð) =
tan(4ð) =
cot(4ð) =
sec(4ð) =
csc(4ð) =
To find the coordinates of point P on the unit circle that corresponds to t = 4π, we can use the fact that on the unit circle, the coordinates of a point (x, y) are given by (cos t, sin t).
1. Coordinates of P:
Since t = 4π, the coordinates of P are (cos (4π), sin (4π)).
2. Trigonometric functions:
Using the values for t = 4π, we can find the exact values of the trigonometric functions.
sin(4π) = sin(0) = 0
cos(4π) = cos(0) = 1
For the remaining trigonometric functions, we can use the identities:
tan θ = sin θ / cos θ
cot θ = 1 / tan θ
sec θ = 1 / cos θ
csc θ = 1 / sin θ
tan(4π) = sin(4π) / cos(4π) = 0 / 1 = 0
cot(4π) = 1 / tan(4π) = 1 / 0 (which is undefined)
sec(4π) = 1 / cos(4π) = 1 / 1 = 1
csc(4π) = 1 / sin(4π) = 1 / 0 (which is undefined)
Therefore, the exact values of the trigonometric functions are:
sin(4π) = 0
cos(4π) = 1
tan(4π) = 0
cot(4π) = undefined
sec(4π) = 1
csc(4π) = undefined
To find the coordinates of point P on the unit circle that corresponds to t, we can use the properties of the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin (0,0) in a coordinate plane. Any point P on the unit circle can be represented as (cos(t), sin(t)), where t is the angle formed between the positive x-axis and the line connecting the origin and point P.
In this case, t = 4π (since 4π = 360°, it represents one full revolution around the unit circle).
To find the coordinates of P, we substitute t = 4π into the equations of the unit circle:
x-coordinate (cosine) = cos(4π) = cos(0) = 1
y-coordinate (sine) = sin(4π) = sin(0) = 0
Therefore, the coordinates of P are (1, 0).
Next, let's find the values of the trigonometric functions of t = 4π.
sin(4π) = sin(0) = 0
cos(4π) = cos(0) = 1
Since we have found the coordinates of P, we can also calculate the values of the other trigonometric functions using the ratios of sides of a right triangle formed with P as one of its vertices.
tan(4π) = sin(4π) / cos(4π) = 0 / 1 = 0 (division by zero does not apply here)
cot(4π) = 1 / tan(4π) = 1 / 0 = NO SOLUTION (division by zero)
sec(4π) = 1 / cos(4π) = 1 / 1 = 1
csc(4π) = 1 / sin(4π) = 1 / 0 = NO SOLUTION (division by zero)
To summarize:
- Coordinates of P: (1, 0)
- sin(4π) = 0
- cos(4π) = 1
- tan(4π) = 0
- cot(4π) has NO SOLUTION (division by zero)
- sec(4π) = 1
- csc(4π) has NO SOLUTION (division by zero)