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.)
9pi/4
Looking for sin, cos, tan,cot, csc, and sec
To find the coordinates of point P on the unit circle that corresponds to t, you can use the trigonometric properties of the unit circle. The unit circle is centered at the origin (0,0) and has a radius of 1.
To find the coordinates of P, use the following formulas:
x-coordinate of P = cos(t)
y-coordinate of P = sin(t)
Let's calculate the coordinates of P for t = 9π/4:
x-coordinate of P = cos(9π/4)
To find the exact value of cos(9π/4), let's first simplify the angle: 9π/4 = 8π/4 + π/4 = 2π + π/4
Now, cos(2π + π/4) = cos(2π)cos(π/4) - sin(2π)sin(π/4)
Remember that cos(2π) = 1 and sin(2π) = 0, so we have:
cos(2π + π/4) = 1 * cos(π/4) - 0 * sin(π/4) = cos(π/4) = 1/√2
Therefore, the x-coordinate of P is 1/√2.
y-coordinate of P = sin(9π/4)
Using the same strategy, we can simplify the angle: 9π/4 = 8π/4 + π/4 = 2π + π/4
Now, sin(2π + π/4) = sin(2π)cos(π/4) + cos(2π)sin(π/4)
Again, cos(2π) = 1 and sin(2π) = 0, so we have:
sin(2π + π/4) = 0 * cos(π/4) + 1 * sin(π/4) = sin(π/4) = 1/√2
Therefore, the y-coordinate of P is 1/√2.
Now let's calculate the values of the trigonometric functions:
sin(t) = sin(9π/4) = 1/√2
cos(t) = cos(9π/4) = 1/√2
tan(t) = sin(t)/cos(t) = (1/√2)/(1/√2) = 1
cot(t) = 1/tan(t) = 1/1 = 1
csc(t) = 1/sin(t) = 1/(1/√2) = √2
sec(t) = 1/cos(t) = 1/(1/√2) = √2
Therefore, the coordinates of P are (1/√2, 1/√2), and the exact values of the trigonometric functions for t = 9π/4 are:
sin(t) = 1/√2
cos(t) = 1/√2
tan(t) = 1
cot(t) = 1
csc(t) = √2
sec(t) = √2