Thursday, February 8, 2007

February 8, 2007 - secant, cosecant, cotangent, arc functions...



Hey guys, sorry for the late post. I suppose. Well, our class today was not much of a ROCKET-SCIENCE (well rocket-math). Well, you know what I mean. It wasn't difficult, YET! First, we started of with the 'BLOG issue'. Have you read the guidelines? Do you agree with them? Are you sure? If not, go ahead and read them and make sure you follow them.
Let's get down to business. What have we learned in this class so far anyway? Let's refresh our memories...

We know that degrees can be converted to radians and Mr. K. has shown us the method of "Proportionality" to perform the mentioned operation. Furthermore, we can convert back and forth from radians to degrees and degrees to radians. The circle above depicts all the radians that will be in most use during the course. They are in multiples of pi or (3.14 ..... ). 'Four corner points' sounds 'weird', because a circle doesn't have corners. They are the points on the unit circle (a circle that has a radius of 1) that intersect with the x-axis and y-axis.


The chart above shows the angles in degrees and in radians, the quadrants they are in, and most importantly the coordinates (x, y) on which they are exactly located on the unit circle. We know that 0 and 360 degrees are on the x- axis on the positive side and 180 degrees is on its opposite side. 90 degrees is on the corner point where it meets the y-axis and 270 degrees on the negative y-axis.

Mr. K. also has taught us a way to remember the exact values and coordinates of each angle (30, 45, 60, and their related angles, 90, 180, and 360). Tan is calculated by the equation:

TAN θ = SIN θ divided by COS θ

... and also has a pattern of:
  • all the radians with denominators of 4, will have 1 as their tangent value, depending on where they are on the cartigean plane.

  • the radians closest to the x-axis will have 1 over root of 3 as their tangent values, and of course, the signs ( + or - ) depends on their location

  • the radians closest to the y-axis will have ROOT of 3 as their tangent values, and of course, the signs ( + or - ) depends on their location

Okay, that is the review. Terms of the day are as follows:

RECIPROCAL - the "flipped" values of a fraction ( ex. the reciprocal of 1 over 2 is 2 over 1 or simply 2 )

SECANT - the reciprocal of cos θ

COSECANT - the reciprocal of sin θ

COTANGENT - the reciprocal of tan θ

INVERSE FUNCTION - it is not the reciprocal but the inverse, it is then calculated using the calculator

So pay attention to this diagram.


The first problem asks: If sin θ is - 2 over 5, and tan is positive, find: cos, csc, sec, tan, and cot θ... AND they are simple to calculate. Just use the length of the sides, set up an equation like the basic ones we used way back in Jr. High for trigonometry ( Soh Cah Toa ). Although this time, once you obtain the values for sine, cosine, and tangent, you have to find their reciprocal values in order to get the values for secant, cosecant, and cotangent.

NOTE: It is okay not to rationalize the denominator!

The second problem asks to solve for x. I KNOW! It is weird because that is quite simple. However, we were just used to punching in values on our calculator and never knew how inaccurate it could be. I wonder why. So, Mr. K. explained how the calculator became unintelligent to the point that it only provides us with one value for sine (for example) instead of 2. We know that sin of 1 over 2 is 30 degrees but wait... as I can recall, our unit circle tells us that there are 2 possible and TRUE sin 1 over 2. They are 30 and 150 degrees, because 150 has a related angle of 30 degrees, which is the only value that the calculator provides us with.

NOTE: arc functions - input of length and output of angles

A couple more problems show the fact that the calculator provides us with insufficient information about the inverse values of sine, cosine, etc. because it only gives us the related angles. Refer to the problems that we took down in class to see more of these arc functions. Note that the quadrants help us determine the exact value for which they indicate whether it is positive or negative. I did not have time to put them all up since this post is already late. Maybe, we all can elaborate from one example and extend that knowledge further to solve similar problems about arc functions and reciprocals. YES, solving for a variable in a normal algebraic equation requires less work and less skills. However, the fact that the calculator is somehow misleading, we have to use our own package of brain cells to determine the exact values.

I hope this post has slightly increased your understanding about what we have learned so far in this course. I apologize for the lateness of my post and my reasons are valid. If I made any mistakes, please feel free to notify me as it can contribute not only to your benefit of learning but also to mine. Thank you. Anyway, to conclude this post, I have to pick the next scribe and she is AICHELLE!



Grey-M said...

Wow! Extremely thorough! Must have taken a good portion of the night. Great scribe.

vincentr said...

Thank you. HAHA! You have no idea how much time I spent on this. I HAVENT EATEN YET. LOL! And it is quarter to nine! I intend to go a little bit overboard.

Mr. Kuropatwa said...

A beautiful scribe post. Simply breathtaking!

Extremely thorough details and beautiful visuals. I think this belongs in the Scribe Post Hall of Fame.

Jojo Rocks said...

Wow, way to set the bar even higher vincent. haha

Tim_MATH_y said...

EXTREMELY AWESOME POST.. wow bar is way too high for the start T.T talk about pressure..

aichelle s. said...

haha wow no kidding! This was beautiful haha, I loved your diagrams. It was amazing. Everything was so elaborate...geez and I'm the next scribe...not that but I'm right after you and my scribe won't even be half as good as yours!!! haha well I'll try!

John D. said...

Vincent, wow, you set the bar absolutely high. Now the scribes in the future will be hard-pressed to top that, not unless they come up with some crazy ideas, haha. Marvelous job! (although i think that was an understatement. lol)

Ricky said...

Excellent Job Vincent. As excepted your scribe is mortally unbelievable. It is better than I would have imagined it to be and more in depth than most of mine. It shows a renowned input of programs and blogger tools and a great knowledge of blogger itself. And you were inducted to the Scribe Post Hall of Fame. Wow... I remember when I was inducted to the scribe post hall of fame... And you set the bar extremely high... You know what that means... One of us is going to try to set the bar even higher. Ha Ha Ha... Later

Lani said...

Hi Vincent,

Your explanations are clear, and the images definitely contribute to understanding the content. Congratulations on autographing your work with excellence!! And your induction into the Hall of Fame!!


VincentR said...

Thanks for the compliments!