Wednesday, February 6, 2008

Circular Funtions and Circles On a Cartesian Plain

Okay I hello everybody, this is benofschool with today's scribe on what we did. We started with talking about Digital Ethics or things to be careful of on the internet. Watch the video on the blog to find out more.

Mr.K also told us what new things to look for on the blog in the near future. Such features include a walkie-talkie sort of thing found on the blog which allows people with a microphone on the computer to send voice messages to other people who are online on the blog. Another cool feature is a little number in the corner of the blog which shows the number of people online which helps with using the walkie-talkie feature. The last but not least feature is a chat/shout box which allows us to chat. But with these features also come trust. If we use these features we must follow the Digital Ethics Rules and keep things appropriate. Remember don't post anything that you don't want anybody to know.

Okay onto the math. We did a small review on what we did yesterday. One reminder about units. There are no units for radian answers. All of the units used in finding the radian answer have been reduced to 1 which doesn't change any answer so it is as if they are not there. One more thing, the equation to convert degrees to radians and vice versa is not a formula but a proportion. It shows how degree values are proportional to radian values. So don't memorize that exact equation but remember how degree values are proportional to radian values.

The first question we worked on is on the second slide on the Feb.6 slide post. One solution is showed on the slide but there was one trick that could be used. We know that a full circle is 2π and we can find out that how the hands are found on a clock at 4:00 that it is one-third of the clock or the full circle. So just divide 2π by 3 which gives us our answer with little work.

The second question we worked on is found on the third slide. (Ignore that little message at the bottom at the slide it is irrelevant). That is the correct solution to the problem and I will now explain the process. To answer the problem you must find the area of the circle using the area of a circle equation. Then make a proportional equation like the one on the slide. It shows that 30° out of 360° is equal to the unknown area out of the total area of the circle. Now all that you have to do is solve for the unknown or on the slide, A. Remember to include units where necessary.

That ended our first class for today. Notice that I said first. There was another one.

In our second class we continued with a new problem and the first question required us to know what the word "coterminal" means. Coterminal means end together. So in the question we need to find out what other radian ends where -π/3 (I don't know how to write fractions on the blog) ends or the positive coterminal of -π/3. So we know that a full circle in radians is 2π. we need to subtract π/3 to 2π to find where it ends. If you are wondering what a negative angle is, a negative angle is the same as going backwards from 360° or 2π. One example would be -50°. That is 360°-50°= 310°. Same goes with radians in this problem. We need to subtract π/3 from 2π. By using basic fraction subtracting skills we can find the answer. To answer the problem we need to multiply 2π by 1 or a number that equals to 1 to change 2π into a number that can help us. We need to multiply 2π by 3/3 and it becomes 6π/3 and now we can subtract π/3.

If you are wondering why we multiplied 2π by 1, we did that to turn the number's appearance in a way that would help us without changing the value of the number. 1 is the identity element of multiplication which means that if we multiply any number by 1 or a number that is equivalent to 1, the value or "identity" of the number doesn't change.

The final problem the class worked with is found on the fifth (5th) slide. P(θ) means the point where a ray intersects the circumference of a circle or in this case the unit circle. The unit circle is a circle that has a radius of one on a Cartesian Plain. To find that angle we can use the SOHCAHTOA rule but we need to find the x and y values of the triangle made shown on the slide. The height or opposite side of the triangle is found by looking at the y-coordinate. The x or adjacent length of the triangle can be found by looking at the x-coordinate the given point. Now that we know two sides we can use the Pythagorean Theorem to find the hypotenuse. But wait, if the sides of the triangle that are not the hypotenuse is 8 units long and 6 units long the hypotenuse must be 10 units long because of the Pythagorean triples. Pythagorean triples are side lengths of triangles are whole numbers like 3, 4, 5 and 65, 72, 97. We can now find the unknown angle made by the ray and the x-axis. Now it doesn't matter where we connect the ray to the x-axis creating a right angle the unknown angle is always that angle found. Even if one of the coordinates were to change integers as in 8 into -8. The angle created by the new created ray with the x-axis will always be that angle. The sine, cosine, and tangent of that angle will be the same as the angle's related angle but in some cases the integer might change depending on which quadrant the angle is found.

Related angles are angles that are the same distance in angles from the x-axis. One example would be 30°. Its related angles would be 150°, 210°, 330° because these values are all 30° or 1/12 away from the x-axis π or 2π.

We can find determine the integer of the sine, cosine, and tangent of an angle by the quadrant in which the angle is located. Sine is a measurement of the y-axis which means in the y-value is the 1st and 2nd quadrant it is positive because the y-value is positive and in the 3rd and 4th quadrant the value of sine is negative as with the negative y-value. For cosine it depends on the integer of the x-value. Tangent is sine/cosine which means that if the sine value is positive and the cosine is negative both according to what was said earlier the tangent would be negative because + / - = -. So cast CAST away and remember the sine=y, cosine=x, and tangent=y/x.

That was all we learned today in class. Homework is to watch the Digital Ethics video, read the Digital Ethics post, Exercise #2, and send Mr.K an email to be invited if you haven't yet.

The next scribe will be nelsa!!!

Bye Bye for now and see you in tomorrow's class. Remember "Beans have souls."

1 comment:

Lani said...

Hi benofschool,

This scribe post is very helpful, especially your explanations: "don't memorize but remember, remember to include units where neccessary, but wait and I'll bet you're wondering" to name just a few.

Nicely done,
Lani