Reference Angle

Reference angles are used to determine the values of the trigonometric functions in the second, third and fourth quadrants, in particular, for the "nice" angles. The reference angle for an angle θ is the smallest angle φ from the (positive or negative) x-axis to the terminal ray of the angle θ.

2nd Quadrant

For an angle θ in the second quadrant the reference angle φ is the remaining angle needed to complete a straight angle, that is, π radians or 180°. Thus θ + φ = π or θ + φ = 180°, and so

φ = π - θ or φ = 180° - θ.





3rd Quadrant

For an angle θ in the third quadrant the reference angle φ is the angle that must be subtracted from θ to leave a straight angle, that is, π radians or 180°. Thus θ - φ = π or θ - φ = 180°, and so

φ = θ - π or φ = θ - 180°





4th Quadrant

For an angle θ in the fourth quadrant the reference angle φ is the remaining angle needed to complete a full circle angle, that is, 2π radians or 360°. Thus θ + φ = 2π or θ + φ = 360°, and so

φ = 2π - θ or φ = 360° - θ.

Signs of sine, cosine and tangent, by Quadrant

The definition of the trigonometric functions cosine and sine in terms the coordinates of points lying on the unit circle tell us the signs of the trigonometric functions in each of the four quadrants, based on the signs of the x and y coordinates in each quadrant.




First Quadrant
For an angle in the first quadrant the point P has positive x and y coordinates. Therefore: In Quadrant I, cos(θ) > 0, sin(θ) > 0 and tan(θ) > 0 (All positive).

2nd Quadrant
For an angle in the second quadrant the point P has negative x coordinate and positive y coordinate. Therefore: In Quadrant II, cos(θ) < 0, sin(θ) > 0 and tan(θ) < 0 (Sine positive).

3rd Quadrant
For an angle in the third quadrant the point P has negative x and y coordinates. Therefore: In Quadrant III, cos(θ) < 0, sin(θ) < 0 and tan(θ) > 0 (Tangent positive).

4th Quadrant
For an angle in the fourth quadrant the point P has positive x coordinate and negative y coordinate. Therefore: In Quadrant IV, cos(θ) > 0, sin(θ) < 0 and tan(θ) < 0 (Cosine positive).

The quadrants in which cosine, sine and tangent are positive are often remembered using a favorite mnemonic.
One example: All Students Take Calculus.

Unit Circle

So... what is a Unit Circle?

A unit circle is a circle with a radius of one (a unit radius). In trigonometry, the unit circle is centered at the origin.

For the point (x,y) in Quadrant I, the lengths x and y become the legs of a right triangle whose hypotenuse is 1.

In the diagram above, we're measuring the angle θ between the x-axis of the Cartesian plane and a line that extends from the origin. Now, here's the really interesting thing; the sine of the angle is equal to the y-coordinate of the point on the unit circle where the line crosses, and the cosine of the angle is equal to the x-coordinate. This is true for any line extending from the origin.

Why is this? Well, the line segment from the origin to the point where it crosses the unit circle forms the hypotenuse of a right-angled triangle. Because the radius of the circle is 1, the length of the hypotenuse is likewise 1. SOHCAHTOA's rules then boil down to:

Sin θ = Opposite

Cos θ = Adjacent

Tan θ = Opposite/Adjacent

In other words:

Sin θ = y

Cos θ = x

Tan θ = y/x

Properties Of Common Tangents

Common Tangent To A Pair Of Circles

Common tangents are lines or segments that are tangent to more than one circle at the same time.

The possibility of common tangents is closely linked to the mutual position of circles.


If two circles touch inside, the two internal tangents vanish and the two external ones become a single tangent.








If two circles intersect, the common tangent is replaced by a common secant, whence there are only two external tangents.









If two circles touch each other outside, the two internal tangents coincide in a common tangent, thus there are three common tangents.










If two circles are separate, there are four common tangents, two inside and two outside.

Tangent Of A Circle.

I started the chapter on circle by introducing the concept of tangent to a circle.

A tangent to a circle has two defining properties

* A tangent touches a circle in exactly one point. This point is called point of tangency.
* The tangent intersects the circle's radius at a 90° angle















Watch Video on Tangent Line


Properties of two tangents to a circle














If two tangents lines are drawn namely AP and BP, they will intersect at a point, P. The properties of the two tangent lines are listed below :

1. CA = CB
2. AP = BP
3. Angle ACP = Angle BCP
4. Angle Angle APC = Angle BPC
5. Angle CAP and angle CBP are both right angles
6. Triangle CAP and BCP are congruent

Blogging Again

It's been quite a while since my last update. It's not that I'm been so busy, it's just that the drive to write was not there. Writing does not come easily for me, I guess its just not my cup of tea.

School reopened on the 1st of September 2009. I managed to finished off the Earth As A Sphere chapter with my 5KA class before they start their trial exam on the 8th of September. I really hope that my girls will do well in this exam. I'm really amazed at the improvements shown by a number of girls in that class. Good Luck to all of you.

My 4SA class did quite well in the August test. Only 2 students were not able to score an A for the test. I'll have to monitor the two girls more closely. I really need to speed up on my teaching coz I just realized that I only have two weeks after the Hari Raya break before the final exam starts. Last week, I missed the Wednesday class coz I had to go for a meeting in Shah Alam. This week I will again miss the Wednesday class due to another meeting in BTP. Thankfully Datin Lee is around to cover for me. I really feel guilty when I had to miss my class but being the ICT coordinator of my school, I'm frequently called for meetings. I wish someone will take over my job coz I'd rather be teaching than doing all the administrative work.

The Blessed Month

Today is the fifth day of the school holidays, so there's nothing to write about school. I hope all the teachers and students are enjoying their holidays. It is really a much needed rest for all of us. The Form 3 students must be working hard coz the PMR exam is coming soon. The Form 5 students also must use the free time wisely to prepare for the trial and SPM examination.

Today is also the fifth day of the fasting month. Starting Ramadhan during the school holiday is really a blessing. I don't need to rush in the morning to go to work. I'm also able to spend more time with my youngest son. This year is the second year he is fasting. Last year he only missed a day of fasting. I'm really proud of him. Hopefully this year he'll be able to complete the whole month of Ramadhan. This year is also the first year he is going to the mosque for the tarawih prayers. He looks forward to go the mosque every night partly because he likes to pray with his friends and partly because of the food that is served after the prayers.

Today I also did a talk on'teknik menjawab soalan' for SPM Mathematics for the children of the staff of the Public Works Department in Kuala Lumpur. This is my first time giving the talk. Overall, it went quite well. The students were very nice. I hope they'll gain something from my talk. As most students and teachers are aware, one of the most effective way to prepare for the SPM examination is to do the past years exam papers. Each year the format and the type of questions are the same. To be good in Math, students need to do a lot of exercises. There's a whole lot of trials exam papers that can be downloaded from the net.

Ramadhan's Coming

The fasting month of Ramadhan is fast approaching. It was confirmed today that the fasting month will start this Saturday. I love the month of Ramadhan. Everything seems so relaxed and peaceful during this month. I don't have to rush like the regular days. I don't have to think about what to eat for breakfast and lunch. I also seem to eat less in this month and thus feel much healthier. Fasting is good for our health. We are able to enjoy the food we have coz we were deprived of it for the whole day. We humans like to take things for granted. We never appreciate the things that we have. Everyone will also make a point to make it home on time to break fast together. Thus the fasting month is also good in strengthening the family ties.

Tomorrow is also the last day of school before the school holidays. The school will be closed for holiday for 1 week from 22 - 31 August. The form 4 classes just finished the August test on Tuesday, so I gave them a day off from lessons on Wednesday. I had to carry on with my lessons with my 5SA class coz the trial exam will start on the 4th of August and I still have a few topics on Earth As A Sphere to cover.

So on Monday, I taught the girls how to find the distance between 2 point on the meridian. These are the steps:

1. Find the difference in angle between the two points.
2. Use the formula Distance = difference in angle x 60 to obtain the distance.

The unit for distance on the sphere is nautical miles (nm). The same steps can be used if you want to find the distance between two points on the equator.

Problem 1 : Find the distance between the following points :

A (85°N , 105°E) and B (27°N, 105°E)

Solution :

Note that points A and B are on the same longitude or the same meridian.

Step 1 : Find the difference in angle between the 2 points. Since the 2 points are in the same hemisphere, substract the latitude of both points to obtain the difference in angle between 2 latitudes.

Difference in angle is (85 - 27) = 58°.

Step 2 : Use the formula Distance = 58 x 60 = 3360 nm.

Problem 2 : Find the distance between the following points

A (30°S , 50°W) and B (15°N , 50°W)

Step 1 : Find the difference in angle. Since the 2 points are in a different hemisphere, we have to add the angles of the two latitudes to obtain the difference between the 2 points.

Thus the difference in angle is (30° + 15°) = 45°

Step 2 : Distance = 45 x 60 = 2700 nm.

The same steps can be use to find the distance between 2 points on the equator.

Example : Find the distance between A (0° , 14°E) and B (0° , 15°W)

1. Difference in angle (14 + 15) = 29 - Different side of the meridian
2. Distance = 29 x 60 = 1740 nm

Schooling On A Saturday

Today is a school day. It is suppose to be a replacement day for the extended holiday the school will be taking for the Deepavali celebration in October. Schooling on a Saturday is not a good idea as proven by the poor attendance of students today. There's only 4 girls present in 5KA and only a handful in my 4SA class today. Merit marks were given to all students who were present today.

My 4SA class was taken over by the physics teacher. The students needs more help in Physics than in Math. As for my 5KA class, I continued with the Earth As A Sphere chapter. Lesson was a breeze today compared with the regular days where I have to handle 23 boisterous girls who doesn't know the meaning of quiet. Its sure is a challenge teaching these girls but I enjoy every minute of it. Today we tried some problems on determining the location of places on the earth's surface.

Any location on Earth is described by two numbers--its latitude and its longitude. If a pilot or a ship's captain wants to specify position on a map, these are the "coordinates" they would use. Actually, these are two angles, measured in degrees, "minutes of arc" and "seconds of arc." These are denoted by the symbols ( °, ', " ) e.g. 35° 43' 9" means an angle of 35 degrees, 43 minutes and 9 seconds (do not confuse this with the notation (', ") for feet and inches!). A degree contains 60 minutes of arc and a minute contains 60 seconds of arc--and you may omit the words "of arc" where the context makes it absolutely clear that these are not units of time.



















To determine the latitude of a location, imagine that the Earth is a transparent sphere (actually the shape is slightly oval; because of the Earth's rotation, its equator bulges out a little). Through the transparent Earth (drawing) we can see its equatorial plane, and its middle the point is O, the center of the Earth.

To specify the latitude of some point P on the surface, draw the radius OP to that point. Then the elevation angle of that point above the equator is its latitude λ--northern latitude if north of the equator, southern (or negative) latitude if south of it. In the diagram below, the latitude of P is 30°N.


















Longitude is distance east or west of a base line called greenwich meridian or prime meridian. The longitude of any given place is its distance, measured in degrees of arc, from this base line.