Chapter 5, Problem 1RCC

(a)

To determine

The equation and definition of unit circle.

Explanation of Solution

A unit circle is a circle with radius one unit whose centre lies on the origin $\left(0,0\right)$.

The equation of a unit circle is shown below.

$x^2+y^2=1$

Therefore, a unit circle is a circle with radius 1 unit whose equation is $x^2+y^2=1$.

(b)

To determine

To explain: The term terminal point determined by t with the help of a diagram.

Explanation of Solution

Consider a unit circle with the starting point $\left(1,0\right)$ and move along the circle t units , the point at which it ends is the terminal point.

Consider a point $\text{P}\left(x,y\right)$ on the unit circle which is reached after covering t distance as shown below.

Figure (1)

It is clearly seen from the above figure that the terminal point is $\text{P}\left(x,y\right)$ after covering distance t.

Therefore, the term used for the end point after covering t distance in counterclockwise direction of a unit circle is terminal point.

(c)

To determine

The terminal point for $t=\frac{\pi }{2}$.

Answer to Problem 1RCC

The terminal point determined by $t=\frac{\pi }{2}$ is $\left(0,1\right)$.

Explanation of Solution

Consider a unit circle with the starting point $\left(1,0\right)$ and move along the circle t units , the point at which it ends is the terminal point. Suppose one-fourth distance in counterclockwise direction is covered on the unit circle. One-fourth distance in terms of radians is $\frac{\pi }{2}$, then the terminal point lies on $\left(0,1\right)$ as shown below.

Figure (2)

Therefore, the terminal point determined by $t=\frac{\pi }{2}$ is $\left(0,1\right)$.

(d)

To determine

To explain: The reference point associated with t.

Explanation of Solution

Let t is a real number.

The reference number $\overline{t}$ which is associated with t is the shortest distance between t and x-axis along a unit circle.

To obtain the reference, it is very important to know the position of t in the quadrant.

If the terminal point lies on first or fourth quadrant, where x is positive then $\overline{t}$ is obtained by moving on the positive x-axis. If the terminal point lies on second or third quadrant, where x is negative then $\overline{t}$ is obtained by moving on the negative x-axis

Thus, the reference number $\overline{t}$ which is associated with t is the shortest distance between t and x-axis along a unit circle.

(e)

To determine

The reference number and terminal point for $t=\frac{7\pi }{4}$.

Answer to Problem 1RCC

The reference number for $t=\frac{7\pi }{4}$ is $\frac{\pi }{6}$ and the terminal point is $\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$.

Explanation of Solution

The reference point $\overline{t}$ is the shortest distance along a unit circle between the terminal point obtained t and x-axis which is associated with t.

The reference number for $t=\frac{7\pi }{4}$ is shown below.

$\begin{array}{c}\overline{t}=\pi -\frac{5\pi }{6}\\ =\frac{6\pi -5\pi }{6}\\ =\frac{\pi }{6}\end{array}$

The reference number is $\frac{\pi }{6}$ which obtains the terminal point as $\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$.

Since the terminal point determined by t is in fourth quadrant therefore, x coordinate is positive and y coordinate is negative.

Hence the terminal point is $\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$.

Thus, the reference number for $t=\frac{7\pi }{4}$ is $\frac{\pi }{6}$ and the terminal point is $\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$.