Why is a “normal” tangent graph uphill, but a “normal” cotangent graph downhill? Use unit circle ratios to explain.
The parent tangent and cotangent graphs are in different directions because their asymptotes are in different places. According to the Ratio Identities, tan=sin/cos and cot=cos/sin. To get an asymptote, the ratio must be undefined (in other words, the denominator has to equal 0.)
How does all that fit in? Tangent has a denominator of cosine, so the places where cosine equals 0 on the Unit Circle and the tangent graph are 90º (pi/2) and 270º (3pi/2). However, cotangent has a denominator of sin, so the places where there would be an asymptote are 0º (0), 180º (pi), and 360º (2pi). Even though the pattern of positive and negatives are the same (+ - + -), because the asymptotes are in different places, the graphs have to be draw differently in order to not touch the asymptotes and follow the rules.
Monday, April 21, 2014
Friday, April 18, 2014
BQ#5: Unit T Concepts 1-3: What's Different About Sine and Cosine?
- Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
Let's start by recapping what the Unit Circle ratios actually are, shall we?
If you will recall, x is the side adjacent to the angle, y is the side opposite to the angle, and r is the hypotenuse of the triangle. In the Unit Circle, r is always equal to 1.
Now notice how sine and cosine are the only trig functions with a denominator of r. Since r always equals 1, sine and cosine will always be real numbers. For the other trig functions, there are places where the denominator in their ratio will equal 0. This makes the value of the trig function in question undefined, and this is where you get an asymptote. Since r will never equal 0, sine and cosine are never going to be undefined and therefore will also never have asymptotes.
Resources:
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-VhUMaI7Gm3UZQdzH9R_WBfFHDd5OS4AKFI9EeCKWTyV6dgymbRguboI59U2nPnlUcgP5hRUcbJovdeBv5C4XjwaxwWhcEHnSS4a9NPEXcupf9LNTOVNPdXJOFZNKooaVZi9uTsclupXr/s1600/trigratios.png
BQ#3: Unit T Concepts 1-3: More Trig Graphs
How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.
1.) Tangent?
Tangent is equal to sine/cosine, according to the Ratio Identity for tangent anyway. An asymptote is created wherever the denominator of a ratio equals 0 (and therefore is undefined), so whenever cosine's y value equals 0 on the graph, there will be an asymptote.
As you can see in this graph of cosine, the points where the y value is zero are -3pi/2, -pi/2, pi/2, and 3pi/2 (there are more, but let's not worry about that.)
In this graph of tangent, the points where the asymptotes are located are -3pi/2, -pi/2, pi/2, and 3pi/2 (again, there are more, but this is just a snapshot.)
2.) Cotangent?
Cotangent is equal to cosine/sine, also according to the Ratio Identity for Tangent. As stated before, an asymptote is found where the denominator equal 0. In this case, the denominator is sine, so the asymptotes are wherever sine's y value equals 0 on the graph.
As you can see in this graph of cosine, the points where the y value is zero are -2pi, -pi, 0, pi, and 2pi.
In this graph of cotangent, the points where the asymptotes are located are 0, 3.14, and 6.28- all of which are (approximately) the same points where sine equals 0.
3.) Secant?
Secant is equal to 1/cos, according to the Reciprocal Identity for secant. (We're done with Ratio Identities for now.) Secant has asymptotes too, which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the secant graph also corresponds with the peaks and valleys of the cosine graph.
4.) Cosecant?
Cosecant is equal to 1/sin, according to the Reciprocal Identity for cosecant. Cosecant also has asymptotes (what a surprise), which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the cosecant graph also corresponds with the peaks and valleys of the sine graph.
Resources:
http://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html
http://www.purplemath.com/modules/triggrph3.htm
1.) Tangent?
Tangent is equal to sine/cosine, according to the Ratio Identity for tangent anyway. An asymptote is created wherever the denominator of a ratio equals 0 (and therefore is undefined), so whenever cosine's y value equals 0 on the graph, there will be an asymptote.
As you can see in this graph of cosine, the points where the y value is zero are -3pi/2, -pi/2, pi/2, and 3pi/2 (there are more, but let's not worry about that.)
In this graph of tangent, the points where the asymptotes are located are -3pi/2, -pi/2, pi/2, and 3pi/2 (again, there are more, but this is just a snapshot.)
2.) Cotangent?
Cotangent is equal to cosine/sine, also according to the Ratio Identity for Tangent. As stated before, an asymptote is found where the denominator equal 0. In this case, the denominator is sine, so the asymptotes are wherever sine's y value equals 0 on the graph.
As you can see in this graph of cosine, the points where the y value is zero are -2pi, -pi, 0, pi, and 2pi.
In this graph of cotangent, the points where the asymptotes are located are 0, 3.14, and 6.28- all of which are (approximately) the same points where sine equals 0.
3.) Secant?
Secant is equal to 1/cos, according to the Reciprocal Identity for secant. (We're done with Ratio Identities for now.) Secant has asymptotes too, which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the secant graph also corresponds with the peaks and valleys of the cosine graph.
4.) Cosecant?
Cosecant is equal to 1/sin, according to the Reciprocal Identity for cosecant. Cosecant also has asymptotes (what a surprise), which are located at the beginning and the end of each period.
Additionally, the minima and/or maxima of the cosecant graph also corresponds with the peaks and valleys of the sine graph.
Resources:
http://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html
http://www.purplemath.com/modules/triggrph3.htm
Wednesday, April 16, 2014
BQ# 2: Unit T Concept Intro: Trig Graphs
How do the trig graphs relate to the Unit Circle?
A trig graph can be seen as an 'unwrapped' Unit Circle. The quadrants of a Unit Circle correspond with the sections on a trig graph. Whether or not the section of the graph is positive or negative -in other words, above or below the x axis- depends on the trig function graphed. For example, since sine is positive in Quadrants I and II and negative in Quadrants III and IV, the pattern of the sections of the trig graph for sine is + + - -, or above the x axis for two sections and below for the next two sections.
Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The period for sine and cosine is 2pi because it takes a length of 2pi along the x axis of the graph to complete the pattern of positives and negatives for sine and cosine. This length is also the length of the circumference of the Unit Circle (how coincidentally convenient!) The period for tangent and cotangent, on the other hand, is pi because it only takes a lengths of pi along the x axis of the graph to complete the pattern of positives and negatives for tangent and cotangent. Since tangent (and cotangent) is positive in Quadrant I, negative in Quadrant II, positive in Quadrant III, and negative in Quadrant IV, it only takes half of the circle to repeat the pattern.
Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig functions don’t have amplitudes) relate to what we know about the Unit Circle?
Sine and cosine have amplitudes because their waves are restricted. On the Unit Circle, sine and cosine cannot be bigger than 1, but neither can they be smaller than -1. The other trig functions don't have these restrictions in the Unit Circle, so they don't have restrictions in their graph either in the form of amplitudes.
A trig graph can be seen as an 'unwrapped' Unit Circle. The quadrants of a Unit Circle correspond with the sections on a trig graph. Whether or not the section of the graph is positive or negative -in other words, above or below the x axis- depends on the trig function graphed. For example, since sine is positive in Quadrants I and II and negative in Quadrants III and IV, the pattern of the sections of the trig graph for sine is + + - -, or above the x axis for two sections and below for the next two sections.
Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The period for sine and cosine is 2pi because it takes a length of 2pi along the x axis of the graph to complete the pattern of positives and negatives for sine and cosine. This length is also the length of the circumference of the Unit Circle (how coincidentally convenient!) The period for tangent and cotangent, on the other hand, is pi because it only takes a lengths of pi along the x axis of the graph to complete the pattern of positives and negatives for tangent and cotangent. Since tangent (and cotangent) is positive in Quadrant I, negative in Quadrant II, positive in Quadrant III, and negative in Quadrant IV, it only takes half of the circle to repeat the pattern.
Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig functions don’t have amplitudes) relate to what we know about the Unit Circle?
Sine and cosine have amplitudes because their waves are restricted. On the Unit Circle, sine and cosine cannot be bigger than 1, but neither can they be smaller than -1. The other trig functions don't have these restrictions in the Unit Circle, so they don't have restrictions in their graph either in the form of amplitudes.
Friday, April 4, 2014
Reflection #1: Unit Q: Verifying Trig Identities
1.) What does it actually mean to verify a trig identity?
To verify a trig identity means to simplify an expression with trig functions until it equals something. When someone asks you to 'verify a trig identity', they mean 'look, here's two expressions that are set equal to each other. Make one side look like the other side.' Usually this consists of simplifying the more complex side to that it has the same terms as the simpler side. It also usually involves substituting trig identities to get things to cancel (and therefore simplify.)
2.) What tips and tricks have you found helpful?
I personally like to convert everything to sine and cosine. It makes it easier for me to see what cancels and what can be substituted (particularly with the Pythagorean Identities.) I also find it helpful to use the Reciprocal Identities to convert everything to the same trig function. Again, it makes it easier for me to see what cancels. Another tip is to try to get things to cancel to 1 (when there's multiplication or division) or 0 (when there's addition or subtraction.)
3.) Explain your thought process and steps you take in verifying a trig identity.Do not use a specific example, but speak in general terms of what you would do no matter what they give you.
Like I explained in the second answer, I like to try to use the Reciprocal Identities to cancel things out. I don't like to convert everything to sine and cosine until later on in the solving process. This is because converting to sine and cosine too early in the solving process can make things needlessly complicated. My last resorts are squaring the expression or multiplying by a conjugate (because those are almost guaranteed to make things very complicated.)
To verify a trig identity means to simplify an expression with trig functions until it equals something. When someone asks you to 'verify a trig identity', they mean 'look, here's two expressions that are set equal to each other. Make one side look like the other side.' Usually this consists of simplifying the more complex side to that it has the same terms as the simpler side. It also usually involves substituting trig identities to get things to cancel (and therefore simplify.)
2.) What tips and tricks have you found helpful?
I personally like to convert everything to sine and cosine. It makes it easier for me to see what cancels and what can be substituted (particularly with the Pythagorean Identities.) I also find it helpful to use the Reciprocal Identities to convert everything to the same trig function. Again, it makes it easier for me to see what cancels. Another tip is to try to get things to cancel to 1 (when there's multiplication or division) or 0 (when there's addition or subtraction.)
3.) Explain your thought process and steps you take in verifying a trig identity.Do not use a specific example, but speak in general terms of what you would do no matter what they give you.
Like I explained in the second answer, I like to try to use the Reciprocal Identities to cancel things out. I don't like to convert everything to sine and cosine until later on in the solving process. This is because converting to sine and cosine too early in the solving process can make things needlessly complicated. My last resorts are squaring the expression or multiplying by a conjugate (because those are almost guaranteed to make things very complicated.)
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