I think that trig identities are equations that take the sin, cos, tan, cot, sec, and csc functions and arranges them into equations using variables, constants, and powers to demonstrate that some variations of these functions are equal to other variations.
And identity is an equation that always works no matter what your input is.
Identity - and equality, trigonometric in this case, that evaluates as TRUE for any value of input; that is both sides of the trig-equations are true for ALL possible variables.
Trig equations that are not identities are conditional equations.
You cannot prove "indentityness" with a graph, but graphs are adequate for disproving it. I suppose what you see is equal, but elsewhere on the graph, there will be places where the the equations aren't if they're not identities.
This was the most interesting class thus far.
Friday, October 16, 2009
Thursday, October 15, 2009
Test
I was sick on Tuesday, and that deprived me of seeing the pretest answer key. That hurt me, but what the heck -- bite the bullet. I think I didn't do too good, but not as bad as the last test. Anyway, I don't feel too choked up over it.
I used to take graphs at face value. Then I discovered a bunch of cool things. Graphs help us visualize numbers. And, with graphs, you can do cool things.
f(z) = x^2 + y^2.
http://www.math.uri.edu/~bkaskosz/flashmo/graph3d/
What does this look like? Well, it's a 3D graph. I tried to visualize it, and it was hard making a three-dimensional leap from the two dimensions, but it sets in once you realize that there are two x^2 parabolas, so, if we twist one of parabolas 90 degrees from the vertex, we create a bowl like structure in 3D:
What does this have to do with what we do in class? This made me realize that the reason they teach us graphs is that these graphs help us visualize the answers to these equations. In the end, these are number solutions that we're looking at. This really puts things in perspective for me.
I used to take graphs at face value. Then I discovered a bunch of cool things. Graphs help us visualize numbers. And, with graphs, you can do cool things.
f(z) = x^2 + y^2.
http://www.math.uri.edu/~bkaskosz/flashmo/graph3d/
What does this look like? Well, it's a 3D graph. I tried to visualize it, and it was hard making a three-dimensional leap from the two dimensions, but it sets in once you realize that there are two x^2 parabolas, so, if we twist one of parabolas 90 degrees from the vertex, we create a bowl like structure in 3D:
What does this have to do with what we do in class? This made me realize that the reason they teach us graphs is that these graphs help us visualize the answers to these equations. In the end, these are number solutions that we're looking at. This really puts things in perspective for me.
Wednesday, October 7, 2009
Reciprocal functions & rational expressions
This post is a gamut of what we did during this week thus far.
On Monday, we did some mental math. Once again, I made some stupid arithmetic errors -- that's something I badly need to improve upon. Ugh.
As for the material we learned in class, it was about transformations using various stretches of functions:
y = a * f(x)
y = f(x * b)
We were also introduced (reminded?) to absolute value functions and their graphs. So, it was about manipulating functions and viewing graphs so we can visualize the solutions to the problem.
If y = abs(x) then we get something that looks like a big V on the Cartesian coordinate plane.
These transformations can be summarized as:
y = a * f(x) | vertical stretch a > 1 | y = 2f(x)
y = a * f(x) | vertical compression 0 < a < 1 | y = 1/2
y = f(b(x)) | horizontal compression b > 1 | y = f(2x)
y = f(b(x)) | horizontal stretch | y = f(1/2x)
On Tuesday, we learned about reflections. No, not the kind that we do on this blog but the kind that pertains to functions.
A function can flip across the x-axis when f(x) becomes -f(x) and f(x) flips across the y-axis when f(-x). We also messed around with inverse functions.
Reciprocals . . . stuff we learned way back in grade five, and stuff we messed around with in grade 9. For example, a/b --------> b/a
so f(x) --------> 1 / f(x)
For inverse functions, the denominator can never be zero (since division by zero is undefined). This is something useful to keep in mind, if we are to try and visualize the function in our head.
Well, that's about it.
On Monday, we did some mental math. Once again, I made some stupid arithmetic errors -- that's something I badly need to improve upon. Ugh.
As for the material we learned in class, it was about transformations using various stretches of functions:
y = a * f(x)
y = f(x * b)
We were also introduced (reminded?) to absolute value functions and their graphs. So, it was about manipulating functions and viewing graphs so we can visualize the solutions to the problem.
If y = abs(x) then we get something that looks like a big V on the Cartesian coordinate plane.
These transformations can be summarized as:
y = a * f(x) | vertical stretch a > 1 | y = 2f(x)
y = a * f(x) | vertical compression 0 < a < 1 | y = 1/2
y = f(b(x)) | horizontal compression b > 1 | y = f(2x)
y = f(b(x)) | horizontal stretch | y = f(1/2x)
On Tuesday, we learned about reflections. No, not the kind that we do on this blog but the kind that pertains to functions.
A function can flip across the x-axis when f(x) becomes -f(x) and f(x) flips across the y-axis when f(-x). We also messed around with inverse functions.
Reciprocals . . . stuff we learned way back in grade five, and stuff we messed around with in grade 9. For example, a/b --------> b/a
so f(x) --------> 1 / f(x)
For inverse functions, the denominator can never be zero (since division by zero is undefined). This is something useful to keep in mind, if we are to try and visualize the function in our head.
Well, that's about it.
Friday, October 2, 2009
Mr. Max talked about the test and success. He believes that a re-test would be great, since he wants to cultivate the idea that we can be successful at this course by tackling the material. I certainly found it to be quite motivational, and I do confess that I haven’t been taking a proactive approach to learning the material in this course. Well . . . I know I did bad on the test – I didn’t learn the material.
Anyway, today’s math topic is transformational geometry. All we are doing is moving and changing graphs around. Altering values in equations can cause translations like this: X^2 + 2
Well, it shifted downwards twice. We did something like this back in grade 10, I believe. One could do the same to trigonometric equations.
Thursday, October 1, 2009
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