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.
2pi
3pi/4
5pi/6
Three questions. I think those are supposed to be t.
To find the coordinates of a point P on the unit circle U that corresponds to a given angle t, we can use the trigonometric functions sine and cosine. The unit circle is a circle centered at the origin (0, 0) with a radius of 1.
1) For t = 2π:
To find the coordinates of P, we need to determine the values of sine and cosine for 2π.
The sine of 2π is 0, and the cosine of 2π is 1.
Hence, the coordinates of P are (1, 0).
The trigonometric functions of 2π are:
sine(2π) = 0
cosine(2π) = 1
2) For t = 3π/4:
To find the coordinates of P, we need to determine the values of sine and cosine for 3π/4.
The sine of 3π/4 is √2/2, and the cosine of 3π/4 is -√2/2 (since cosine is negative in the second quadrant).
Hence, the coordinates of P are (-√2/2, √2/2).
The trigonometric functions of 3π/4 are:
sine(3π/4) = √2/2
cosine(3π/4) = -√2/2
3) For t = 5π/6:
To find the coordinates of P, we need to determine the values of sine and cosine for 5π/6.
The sine of 5π/6 is 1/2, and the cosine of 5π/6 is -√3/2 (since cosine is negative in the second quadrant).
Hence, the coordinates of P are (-√3/2, 1/2).
The trigonometric functions of 5π/6 are:
sine(5π/6) = 1/2
cosine(5π/6) = -√3/2
By using the unit circle and applying the definitions of sine and cosine, we can find the coordinates of P and the exact values of the trigonometric functions for different angles.