Question

Find the terminal point P(x, y) on the unit circle determined by the given value of t.

t=3π/2Find the reference number for each value of t.
(a) t= 13π/6
(b) t= -9π/7
(c) t = 6
(d) t = -7

Answer 1

The terminal point for t = 3π/2 on the unit circle is P(0, -1), and the reference numbers for each given value of t are π/6, 5π/7, 6 - 2π, and 2π - 7, respectively.

Step-by-step explanation:

To find the terminal point P(x, y) on the unit circle for a given value of t, we look at the angle t corresponds to on the unit circle. For t = 3π/2, this is the angle that points directly down from the centre of the circle. The coordinates for this point are P(0, -1).

To find the reference number for each given value of t, we determine the corresponding angle in the first revolution (0 to 2π) that has the same sine, cosine, and tangent values as t.

1. t = 13π/6: By subtracting full revolutions (2π), we get the reference number as π/6.
2. t = -9π/7: By adding revolutions until the angle is positive and within the first revolution, the reference number is 5π/7.
3. t = 6: 6 radians is just slightly less than a full revolution, so the reference number is 6 - 2π.
4. t = -7: -7 radians would correspond to an angle just over a full negative revolution, and the reference number is 2π - 7 (since we add 2π to make it positive).