**How To Find Terminal Point** – 2 The Unit Circle In this section we discuss some properties of a circle of radius 1 centered at the origin. The set of points at a distance 1 from the origin is a circle of radius 1 (see Figure 1). Unit Circle Figure 1

Let t be a real number. If t 0, let t denote the distance in a unit circle starting at point (1, 0) and moving counterclockwise. If t 0 (b) Terminal point P(x, y) determined by t < 0 Figure 2

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## How To Find Terminal Point

Thus we arrive at the point P(x, y) on the unit circle. The point P(x, y) thus obtained is called the terminal point determined by the real number t. Circumference of unit circle C = 2(1) = 2. So if a point starts at (1, 0) and moves counterclockwise in a unit circle back to (1, 0), it travels a distance of 2. To move to the center of the circle, it travels a distance of (2) = . To move a quarter of the distance around the circle, it covers a distance of (2) = /2.

### Solved: Find The Values Of The Trigonometric Functions Of T From The Given Information. 42 Sin(t) Terminal Point Of T Is In Quadrant Iv 13 Cos(t) 55/12 Tan(t) 412/5 Csc(t) 412/13 Sec(t)

Where does the point end up when these distances are traveled around the circle? For example, when it starts at (1, 0) and travels a distance of , we see in Figure 3 that its terminal point is (–1, 0). t = , , , and 2 Figure 3 specified terminal points

Find the coordinates of the terminal point on the unit circle determined by each real number t. (a) t = 3 (b) t = – (c) t = –

Similar methods can be used to find the terminal points specified by t = /6 and t = /3. Table 1 and figure 6 give the terminal points for some special values of t. Table 1 Figure 6

Using symmetry, find the terminal point determined by each real number t. (a) t = (b) t = (c) t =

#### Find The Terminal Point Q Of The Vector V With Components As

13 Reference Number To find a terminal point in any quadrant, we only need to know the “corresponding” terminal point in the first quadrant. We use the idea of reference number to help us find the terminal points.

14 Reference Number Figure 8 shows that to find the reference number t, it is helpful to know in which quadrant the terminal point specified by t is located. t Reference No. 8. Figure

Find the reference number for each value of t. (a) t = (b) t = (c) t = (d) t = 5.80

Using reference numbers, find the terminal point determined by each real number t. (a) t = (b) t = (c) t =

## Solved: 1) Given A 3 Dimensional A = (2, 3, 4). Find Its Magnitude 2) Given The Vector A Has An Initial Point At The Origin And A Terminal Point At (3, 2)

18 Reference Number Since the circumference of a unit circle is 2, the terminal point determined by t is the same as that determined by t + 2 or t – 2. In general, we can add or subtract 2 any number of times without changing the terminal point specified by t. We use this observation in the next example to find the terminal points for large t.

In order for this website to function, we record user data and share it with processors. To use this website, you must accept our privacy policy, including our cookie policy.2 The Unit Circle In this section we will analyze some properties of a circle of radius 1 centered at the origin. The set of points at a distance 1 from the origin is a circle of radius 1 (see Figure 1). Unit Circle Figure 1

Show that a point P lies on the unit circle. Solution: We have to show that this point satisfies the equation of the unit circle, i.e. x2 + y2 = 1. Since P lies on the unit circle.

Let t be a real number. If t 0, let t denote the distance in a unit circle starting at point (1, 0) and moving counterclockwise. If t 0 (b) Terminal point P(x, y) determined by t < 0 Figure 2

#### Solved] A Terminal Point, P (x, Y), On The Unit Circle Forms An Angle In…

Thus we arrive at the point P(x, y) on the unit circle. The point P(x, y) thus obtained is called the terminal point determined by the real number t. Circumference of unit circle C = 2(1) = 2. So if a point starts at (1, 0) and moves counterclockwise in a unit circle back to (1, 0), it travels a distance of 2. To move to the center of the circle, it travels a distance of (2) = . To move a quarter of the distance around the circle, it covers a distance of (2) = /2.

Where does the point end up when these distances are traveled around the circle? For example, when it starts at (1, 0) and travels a distance of , we see in Figure 3 that its terminal point is (–1, 0). t = , , , and 2 Figure 3 specified terminal points

Find the terminal point of the unit circle defined by each real number t. (a) t = 3 (b) t = – (c) t = – Solution: From Figure 4 we get: Figure 4

9 Example 2 – Solution Continue (a) The terminal point determined by 3 is (–1, 0). (b) The terminal point determined by – is (–1, 0). (c) Terminal point (0, –1) determined by – /2. Note that different values of t can determine the same terminal point.

## Solved:51 54 Use The Figure To Find The Terminal Point Determined By The Real Number T, With Coordinates Correct To One Decimal Place. T= 1.1

The terminal point P(x, y) determined by t = /4 is the same distance (1, 0) along the unit circle (see Figure 5). Figure 5

Since the unit circle is symmetrical about the line y = x, P lies on the line y = x. Therefore, P is the point of intersection of the circle x2 + y2 = 1 and the line y = x (in quadrant I). Substituting y into the equation of the circle for x, we get x2 + x2 = 1 2×2 = 1 combining terms.

Since x2 = x = P is in quadrant I, since x = 1/ and y = x, we also have y = 1/. So divide the terminal point determined by /4 by 2 take square roots

Similar methods can be used to find the terminal points specified by t = /6 and t = /3. Table 1 and figure 6 give the terminal points for some special values of t. Table 1 Figure 6

### Answered: Find The Values Of The Trigonometric…

Find the terminal point defined by each real number t. (a) t = (b) t = (c) t = Solution: (a) Let P be the terminal point determined by – /4 and Q be the terminal point determined by /4.

15 Example 3 – Solution Forward From figure 7(a), we see that P has the same coordinates as Q except for the sign. Since P is in quadrant IV, its x-coordinate is positive and its y-coordinate is negative. So the terminal point is P( /2, – /2). Figure 7 (a)

16 Example 3 – Solution continued (b) Let P be a terminal point determined by 3 /4 and let Q be a terminal point determined by /4. From Figure 7(b) we see that point P has the same coordinates as Q except for the symbol. Since P is in quadrant II, its x-coordinate is negative and its y-coordinate is positive. So the terminal point is P(– /2, /2). Figure 7 (b)

17 Example 3 – Solution continued (c) Let P be the terminal point determined by –5 /6 and Q be the terminal point determined by /6. In Figure 7(c), we see that point P has the same coordinates as Q except for the sign. Since P is in quadrant III, both its coordinates are negative. So the terminal point P in figure 7 (c)

#### Solved] Vector Has Initial Point P (2, 14) And Terminal Point Q…

18 Hint No. From examples 2 and 3 to find a terminal point in any quadrant we only need to know the “corresponding” terminal point in the first quadrant. We use the idea of reference number to help us find the terminal points.

19 Reference Number Figure 8 shows that to find the reference number t, it is helpful to know in which quadrant the terminal point denoted by t is located. t Reference No. 8. Figure

20 If the reference number terminal point is in quadrant I or IV, if x is positive, we find t by moving along the circle toward the positive x-axis. If it is in quadrant II or III, if x is negative, we find t by going around the circle

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