00:02 My name is John Ogbodo.

00:12 I'll be looking at today,

00:14 what's the called trigonometry, ratios of special angles

00:18 and the topic is, or the subject is photo mathematics.

00:22 So under this, we are going to understand

00:24 and look at the basic concepts of trigonometry.

00:28 And under this, the major objective here

00:31 are the measurement of angles

00:34 and the meaning of angles, trigonometrical functions

00:37 and ratios, reciprocals of angles and functions

00:41 and triangular coordinate and sign convention

00:44 and what we call trigonometrical ratio.

00:48 We also look at, by extension,

00:50 what we call the Pythagoras theorem

00:54 and what we call the Sokatoa.

00:58 So what's this trig?

01:00 Basically, the word trig came from the Greek word

01:03 and the Greek word means measurements of triangles

01:08 or measurement of angles.

01:09 Trig is a branch of mathematics that is concerned

01:12 with the measurement of sites and angles of plane triangle

01:15 and the investigation of the various relations

01:18 which exist among them.

01:20 The importance of trigonometry is very important

01:24 in such a way that's easy to solve problems in measurement,

01:30 mechanics in physics, solving and astronomy.

01:33 So trig is a very wide subject.

01:36 Now, two major aspects of trigonometry

01:39 are plane, trig and spherical trigonometry.

01:42 And plane trigonometry is concerned

01:44 with angles, triangles and other figures

01:47 which lie on the plane.

01:49 And spherical trigonometry is concerned

01:52 with the spherical triangles that is triangles

01:55 that lie on the sphere or circle,

01:58 size of which, such as the circular acts

02:02 and the conical shapes of circular extensions.

02:06 So here we move on to what an angle is.

02:10 So what is an angle, basically?

02:12 An angle is defined as the union of two non-colonial rays,

02:15 which have a common end points.

02:18 An angle is also defined as a measure of the rotation of line

02:20 from one position to another about the fixed point on it.

02:24 So we look at the term intercession of two lines.

02:30 Let's call this line two and line one.

02:34 When the two of them intersect at points o,

02:37 they form an angle.

02:38 This angle is called angle theta

02:41 or any other variable that we limit fit to name it.

02:44 So angles are measured by using certain protractors

02:50 and devices that can measure the angular acts.

02:54 And angles can be measured in degrees or in radiance.

02:58 So basically here, we are going to be considering

03:00 the different types of angles.

03:02 We have the positive angle and the negative angle.

03:05 If the angle lies above the x-axis,

03:09 it becomes a positive angle.

03:11 And if it lies below the x-axis, such as the following,

03:16 it becomes a negative angle.

03:18 So the reference point is the x-axis.

03:22 The x-axis is the reference line or reference plane

03:27 when we are measuring angle.

03:29 So here, we also move on to a term called the quadrant.

03:33 What is quadrant?

03:34 The quadrant is a concept that we use to divide the angle

03:40 into various aspects of dimension,

03:43 such as when we divide the seco.

03:45 We look at the seco.

03:46 Remember, the seco is the local's of points

03:49 that I could distance from a fixed point called the center.

03:52 And this equidistant points are always known

03:56 as what's called the radius.

03:57 So we have this.

03:58 This is the center.

04:00 From here to here is radius.

04:01 From here to here is radius.

04:03 Radius, radius, radius, radius, up to infinity time.

04:07 So there's a constant distance from the center

04:11 o to the circumference.

04:15 That's constant distance is called the radius.

04:18 So when we divide a seco into four equal parts, this is a seco.

04:24 It will draw a line and another line dividing into four equal sides.

04:27 We have o to y.

04:30 If you consider this vertical as the y-axis and o to x as the x-axis.

04:36 So you can also have the negative x-axis and the negative y-axis.

04:41 So with this, we call this the quadrants.

04:45 Four quadrants of the seco.

04:47 We have the first quadrant.

04:49 We have the second quadrant.

04:51 We have the third quadrant and the fourth quadrant.

04:55 So we have four quadrants, which we need to understand

04:59 and how it is applied as we move along.

05:02 So here, we look at the, in the first quadrant,

05:05 how does the angles behave?

05:08 Well, look at the first quadrant, how does the angles behave?

05:11 Now, in that case, we consider the sine, cosine,

05:16 and tangent of angles.

05:19 From this quadrant, we call this zero.

05:23 We call this 90.

05:24 We call this 180 degrees.

05:27 And here is 270 and back to 360 degrees.

05:31 So a full seco is 360 degrees.

05:34 So from 0 to 90 and 90 to 180, 180 to 270 and the 270 to 360.

05:47 So 0 to 90 is the first quadrant, OK?

05:50 That's the first quadrant.

05:52 90 to 180 is the second quadrant, that's the second quadrant.

05:57 180 to 270 is the third quadrant.

06:01 And 270 to 360 is the fourth quadrant.

06:06 So we have four quadrants in the seco.

06:08 So with this, we have divided the seco into four quadrants.

06:12 And without four quadrants, we can know

06:15 where a value of sine, cosine, or tangent

06:18 can be positive or negative based on the values

06:22 of the angles given.

06:24 OK.

06:25 How do we measure angles?

06:28 The size of any angle is determined by the amount of rotation.

06:31 In trig two, system of measurement are considered.

06:33 We have the exact gessimal or English system

06:36 called the degree measurement.

06:38 And we have the circular measure system, which

06:40 is called the radians.

06:42 We are going to look at the radians with them.

06:44 But let's consider the degree measurement.

06:47 In the degree measurement, we talk about this system.

06:51 Where measurement of angle is in degree.

06:53 And the major aspect of it is that the seco is

06:57 very into 90 degrees.

07:00 That is four equal angles of a seco.

07:04 And for the circular measure system,

07:06 which is called the radians system,

07:08 in this system, an angle is measured in radians.

07:10 In higher mathematics or more advanced maths,

07:13 we talk about this, usually, in circular system.

07:17 And it is very important when we are measuring angles

07:21 in radians.

07:22 OK.

07:22 So that is for the angles.

07:26 So let's come back to the concept of radian

07:30 and degree relationship, radians and degree relationship.

07:34 What is the relationship between the radian and angle

07:36 and the degree angle?

07:37 Radian and degree relationship.

07:40 Now, the radian and the degree relationship.

07:45 What is the relationship?

07:47 Let's see how we can relate them and do

07:51 conversion from one form of measurement to the other.

07:55 So we have the radian angle, the 360 degrees for a full cycle

07:59 is equal to 2 pi, 2 pi radian.

08:03 So the 2 pi radian is 360 degrees.

08:06 So the radian and the degree angles

08:09 can be related by using the relationship 2 pi radians

08:13 equal to 360 degrees.

08:15 So with that, we can also find on our relationship

08:17 like smaller angles.

08:18 When we have 180, if we develop a side by 2,

08:22 we have 2 pi divided by 2, 360 degrees divided by 2.

08:28 So 2 consists of 3.

08:29 So pi radian is equal to 180 degrees.

08:34 Pi radian is 180 degrees.

08:37 Pi over 2 radian, pi over 2 radian is 90 degrees.

08:42 And if we divide pi by 3, we have pi over 3 radian

08:48 is equal to 180 over 3.

08:53 And 180 over 3 is equal to 60 degrees.

08:58 So pi over 3 radian is equal to 360 degrees.

09:02 What about other relationship?

09:05 Let me say we want to look for 15 degrees in radian.

09:09 15 degrees in radian.

09:10 How do you convert 15 degrees in radian?

09:13 15 degrees in radian, x radian.

09:18 What do we do?

09:18 We find the relationship.

09:20 In terms of pi and radian, we record that pi

09:24 is equal to 180 degrees.

09:27 So when we cross multiply, we have pi is 180.

09:32 Or we say 15 degrees equal to x radian.

09:38 And pi, that means pi radian is equal to 180 degrees.

09:45 So when we cross multiply, we have 15.

09:49 Pi equal to 180 x.

09:55 We're divided by 180.

09:58 Since we are converting to radian,

10:00 we're divided by 180.

10:02 So 180, 180, 180 comes to 180.

10:06 And then we can divide this by 3.

10:10 Let's say by 5, 5 here, 3, 5 here.

10:14 We have 5 in this, we have 3.

10:17 Remember, 3 in 36.

10:20 3 can be divided by 3 here.

10:22 And 3 here is 12.

10:24 So that means pi over 12 is equal to x.

10:29 We are x equal to 15 degrees.

10:31 So that means 15 degrees is equal to pi over 12.

10:36 So we can get all the angles converting from one form or one

10:42 measurement to the other.

10:43 So let's convert 20 degrees.

10:45 Let's convert 20 degrees to radians.

10:50 To x radian, I also convert pi over 8 radians to degree.

10:58 To-- let me call it y degree.

11:00 So we call this 1.

11:01 We call this 2.

11:02 So how do we do this?

11:04 Quickly, we say that 20 degrees to--

11:07 we recall that pi radian equal to 180 degrees.

11:16 Pi radian equal to 180 degrees.

11:17 So therefore, if we want to convert 20 degrees to radian.

11:23 So we put x.

11:24 What we do is cross multiplication.

11:27 We have 25 equal to 180 x.

11:31 So we multiply by 180, 180.

11:36 So this comes to this, 0 comes to 0, 2 here, 1, 2 here, 9.

11:41 So x equal to pi over 9 radians.

11:47 So that means 20 degrees is equal to pi over 9 radians.

11:54 And y degrees equal to pi over 8.

11:56 What is y?

11:58 If we want to solve this, we also apply the same concept

12:01 to do that we have, again, pi radian equal to 180 degrees.

12:09 And pi over 8 pi over 8 radian equal to x or y.

12:16 Let's put y.

12:17 So what we cross multiply again, we have pi x--

12:21 that is pi y-- equal to pi over 8 multiplied by 180.

12:27 So it can-- it can see, pi can see pi both sides.

12:33 2 here, we have 4.

12:36 And 2 here, we have 90.

12:40 And again, we can go with 2 here, we have 2.

12:44 And 2 here, we have 45.

12:47 And we again, 2 here, 1.

12:49 And 2 here, we have 22.5 or 22, 1 over 2.

12:55 So that means pi over 8.

13:00 Look as a pi over 8 equal to 22, 1 over 2 degrees.

13:10 That is a value.

13:11 So that is a easy conversion from 1 degree

13:15 or from degree to radian and from radian to degree.

13:21 But in addition to that, we need to understand

13:24 that there are other conversion, we can also

13:27 come across where we talk about degree and minute.

13:30 Remember that we can subdivide 60 equal parts into second.

13:34 Remember, there are 1 minute, 60 seconds.

13:36 And 1 degree is 60 minutes.

13:39 That means 1 minute.

13:41 1 minute is 1 minute equal to 60 seconds in measurement.

13:49 And 1 degree is equal to 60 minutes.

13:57 1 degree is 60 minutes.

13:59 And this 60 minute is sentinel.

14:01 So 60 multiplied by 60 seconds.

14:04 And 60 multiplied by 60 is 3,600 seconds.

14:09 So that means 1 degree.

14:11 1 degree equal to 3,600 seconds.

14:17 And degree in angles can be represented as a sign.

14:21 Because sometimes we have to convert some sudden

14:24 this measurement.

14:25 Maybe you have something like a 60 degrees, 60 degrees

14:29 and the 2 minutes.

14:32 60 degrees and 2 minutes is the sentinel.

14:35 So in 60, 1 is 2 over 60, which is sentinel as 60, 1 over 30 degrees.

14:46 This is how to express it.

14:47 And if it is 60 degrees and the 30 minutes, this minute, 30 minutes.

14:56 It's sentinel sends 60 degrees, 30 divided by 60.

15:02 So that means we have 61 over 2 degrees.

15:07 61 over 2.

15:08 Because 30 cancels this, we have 1 over 2.

15:11 So that is a way of converting from minutes to degree.

15:16 And that is their relationship.

15:17 Now, we move ahead to other aspects

15:19 where we talk about the circular measure,

15:22 which we have also seen the relationship.

15:24 Now, talking about the circular measure, which is the radian angle,

15:27 we said we can convert from radian to degrees

15:30 and from degrees to radian.

15:32 So we can also look at other related terms,

15:35 where we actually need to apply them

15:38 and solve some related terms in terms of radians and degrees.

15:42 So we have this convert, 37 degrees to 7 minutes, 38 seconds,

15:47 into a radian measure.

15:49 For it to do that, we need to find the relationship

15:52 between each of them, which we have stated there.

15:55 And let's see how we can do a related example with respect to that.

16:00 So let's consider that.

16:02 We have to convert, let me say, 20 degrees, 20 degrees, 30 minutes,

16:11 and 20 seconds, into radian angle.

16:17 Now, first of all, we have 20 degrees, 20 degrees,

16:22 and 30 over 60, and 30 over 60, and 20 over 3,600.

16:32 And if we have that, we divide this, and we divide this, too.

16:36 So this is interesting, two here, we have over 180.

16:39 So that means we have 20 degrees, 1 over 2, 1 over 2, comma, 1 over 180.

16:50 So with that, you can simplify further and convert to radian.

16:55 The major angle here is the 20 degrees.

16:57 And to convert 20 degrees to radian, we have your pile is equal to 180 degrees.

17:07 Therefore, 20 degrees is converted to x.

17:10 So what we do is cross multiplication.

17:12 We have 20 pile equal to 180 x.

17:18 So what we'll divide by 180.

17:21 We will say that 180, this counts to this, 0 counts to 0.

17:25 So we have x is equal to 1 over 9.

17:31 So x equal to 1 over 9, pi, or we can say pi divided by 9 radians.

17:39 So that is how to convert this.

17:40 So with the half and where 180, the approximated value,

17:44 which is equal to 1 over 3,000, or 1 over 360 degrees.

17:50 So 1 over 360 would be a very small angle in conversion.

17:55 So therefore, what you do is you run it off to 1 radian approximately.

17:59 So therefore, 1 over 360 degrees, or degrees in this form,

18:07 can be converted to radian x by using the same relationship.

18:13 And what we do there is that we say 1 over this, which is we know that pi equal to 180.

18:18 So 1 over this is-- so we do a cross multiplication by swapping it.

18:23 So what we do the conversion, we can now see the values.

18:27 For example, what we are having here is 180 degrees is equal to pi.

18:33 And the 1 over 360 degrees is equal to x.

18:40 So what we do are cross multiplication.

18:42 We have 180 x equal to 1 over 360, all pi.

18:50 So we divided by 180, that was a by 180.

18:54 So this comes to this.

18:56 And when that comes to this, this will now become x equal to 1 over 360 degrees,

19:04 360 multiplied by 180, all pi.

19:08 So you can do the multiplication to that.

19:10 And get the answer for the value of x.

19:13 So with that, we know that that is for the circular measure.

19:18 Now, in our aspects, we need to look at this.

19:20 How to deal with the arc, the length of arc in a seco, in terms of radians?

19:27 We consider a seco.

19:29 And this seco have radios and angle subtinality center.

19:33 These radios are radios.

19:35 The length is L. The length of arc in degrees

19:38 is length equal to theta over 360 degrees, multiplied by 2 pi r.

19:47 That is the length of arc.

19:48 But the length of arc, when you are dealing with radians,

19:52 since we know that 2 pi is equal to 360 degrees,

19:58 we can substitute into that.

20:00 We have length equal to theta over 2 pi multiplied by 2 pi radios.

20:07 So our angle is now measured in radians.

20:11 So 2 pi, 2 pi.

20:13 So length equal to theta r.

20:17 So length equal to theta r. Where theta is in radians?

20:23 And our theta can be equal to the length divided by r.

20:27 The ratio of length to the radius.

20:31 We have the-- and the radius can also be calculated or determined,

20:34 by using length over the theta in radians.

20:38 So we take note of these two special angles.

20:43 OK, so if you are giving an example such

20:46 that the angle subtended at the center is in radians,

20:52 let's consider this sector of a seco.

20:56 The angle subtended here is in radians.

20:58 Let's say it is a 4 pi.

21:01 If it's 4 pi, we can find this length if radios

21:03 is given as 1 meter.

21:06 If radios is 1 meter, the length is equal to radios multiplied

21:12 by theta.

21:13 So that means radios is 1 meter, and theta is 4 pi.

21:17 So the answer there will give us 4 pi meter.

21:22 That is the length of the arc in terms of pi.

21:25 But remember that your theta here is in radians.

21:28 Theta equal to 4 pi.

21:30 OK, so with that, we move on to the subsequent

21:33 one, where we look at all that relationship

21:36 in terms of length of circular arc, which we have done,

21:40 and the relation in terms of speed and the likes.

21:45 OK, so looking at example, we say, what is the length of arc

21:48 of radios?

21:50 First, as a whose center angle is 140 degrees?

21:53 So we can also do the conversion from 140 degrees

21:57 to radians, or we do a straight calculation

22:00 using the degree measurement.

22:02 So we can as well do that in degrees.

22:07 Your example says a core of on a highway is laid out

22:11 out of an arc of a circle of radios 620 meter.

22:16 How long is the arc that's obtained a center angle of 32 degrees?

22:20 Now, for us to do or deal with this,

22:23 we can solve the question where we are looking

22:26 at the length of the arc, the length of the arc.

22:31 That's obtained the angle of 32 degrees.

22:33 Now, the length of the arc that's obtained an angle of 32 degrees,

22:36 we have the radios.

22:39 Radios are 620 meters, and the angle

22:43 is obtained at 32 degrees.

22:46 Now, if you do that, we have length equal to theta

22:49 over 360 degrees, multiplied by 2 pi r.

22:54 So we have the theta as 32.

22:57 And the divided by 360 degrees times 2 pi radius.

23:05 Radios is 620 meters.

23:09 We can convert 360 degrees to radians.

23:11 We have 32 divided by 2 pi times 2 pi, then times 620.

23:21 So with this, this can do this.

23:22 So multiply 32 by this.

23:25 And our final answer will be 346.41 meters.

23:32 Well, we get the answer in radians.

23:33 So we multiply this, converting to radians.

23:37 Our final answer for the length will now be 346.41 meters.

23:42 So with this, we can also consider some other relationship

23:48 in terms of railway traveling on a curve

23:51 of half a kilometer radios at the rate of 20 kilometer power,

23:55 through what angle had it turned in 10 seconds.

23:58 So through what angle had it turned?

24:00 How is it going to turn through the angle?

24:02 So we are considering a curve path.

24:05 A curve path where it really is traveling.

24:09 Half a kilometer would give us half kilometer

24:13 is sent to us half times 1,000.

24:16 Half times 1,000 is 500 meters.

24:20 So 500 meters.

24:22 And this speed is velocity, which is speed V,

24:27 is 20 kilometer per hour, 20 kilometer per hour.

24:33 And they said, how long?

24:36 If we have gotten times seconds, we have the times 10 seconds.

24:40 And we are looking at the distance

24:43 that the vehicle or the train has traveled.

24:46 So together, we multiply this distance

24:49 and look at the other relationship.

24:51 We have the distance, cover distance,

24:54 is equal to velocity multiplied by time.

24:59 Velocity multiplied by time.

25:00 Now, since it has covered this distance,

25:03 in this velocity distance in meters,

25:07 velocity is 20 kilometer per hour.

25:09 So let's multiply the half kilometer cover.

25:13 We don't be 20 multiplied by half.

25:17 And that is 10.

25:19 10 kilometer within the short time.

25:22 So therefore, the total distance cover

25:26 will not be equal to 10 multiplied by 10 seconds

25:31 divide, convert to hours.

25:34 That would be 10 over 3,600.

25:38 So we have 100 divided by 3,600.

25:44 So 0,000.

25:46 So the final answer is 1 over 36 kilometer within that space

25:52 of time.

25:53 So that is the answer for that, which

25:55 respect to the radian angular measurement.

25:59 So we have other relation we talk about arc

26:02 and the sector relationship.

26:04 And when we do that, we apply the same concept

26:08 in solving it using the concept of radian and degree

26:13 relations.

26:13 So we have the following exercises we can solve.

26:17 Convert the following radian measure to degrees

26:20 and convert degrees to radian.

26:22 We have 2, 10 degrees to radian.

26:24 540.

26:26 We have 42 degrees, 32 minutes, 12 seconds.

26:29 And the 24 degrees, 32 minutes, 30 seconds.

26:33 And we have 5 pi over 4, 2 pi over 3, 5.52 radians,

26:40 and 1.3 is a radian.

26:42 Another related examples, which we can also solve using

26:46 what we have learned.

26:48 So we have the answers, which we have also provided at the end.

26:52 So we want to appreciate you for watching the video.

26:57 And we thank you for watching.

26:59 My name was again is John Ubudu.