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Tuesday, April 22, 2014

BQ #4: Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle to explain.

The tangent graph is uphill and the cotangent graph is downhill because of the asymptotes and they are different because of the trig ratios. Tangent has a ration of y/x and cotangent is x/y. Asymptotes exist when the trig function is undefined and the denominator equals zero. Tangent is undefined when it is at pi/2 and 3pi/2 which is where x=0. Cotangent is undefined when y=0 and on the unit circle it would be at 0 and pi.
The signs are the same for both graphs in the quadrants which is positive, positive, negative, negative. The graphs are different because the asymptotes are in different places and the graphs are shifted and that makes tangent go uphill and cotangent go downhill because of the signs.









Monday, April 21, 2014

BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each other? Emphasize asymptotes in your response.

A. Tangent? The graph of tangent has asymptotes based on cosine and you need to know that the asymptotes are found. The ratio for tangent is sine/cosine so in a graph where the value is undefined or when cosine is equal to zero. So if cosine equals zero, tangent is undefined and it has asymptotes. Cosine equals pi/2 and 3pi/2, so we know where that the asymptotes lie there.

B. Cotangent? The graph's asymptotes depend on sine and the ratio is cosine/sine. We know that it is undefined when the denominator is equal to zero which is the asymptotes. As a result, when sine(x)= 0, cotangent is undefined. Sine equals zero at 0 and pi, which is where the asymptotes lie.

C. Cosecant? The asymptotes are based on sine since it is the reciprocal and the ratio is 1/sine. We know that the denominator has to be equal to zero which would give us the asymptotes. If sine(x)=0, cosecant is then undefined and we have asymptotes. Sine equals zero at 0 and pi, which is where the asymptotes lie. Also, the positive and negative values are dependent on sine because they share the same ones.

D. Secant? The graph of secant has asymptotes that depend on cosine it is the reciprocal and the ratio is 1/cosine. We know that the denominator has to be equal to zero which would give us the asymptotes. If cosine(x)=0, cosecant is then undefined and we have asymptotes. Sine equals zero at pi/2 and 3pi/2, which is where the asymptotes lie. Also, the positive and negative values are dependent on cosine because they share the same ones.

Thursday, April 17, 2014

BQ #5: Unit T- Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain.
Asymptotes are found when a ratio is divided by zero, or it is undefined. Sine y/r and cosine x/r are never undefined because they are both divided by 'r', where on the Unit Circle 'r' equals 1. However, cosecant r/y, secant r/x, tangent y/x, and cotangent x/y would have asymptotes. The fact that they have y or x as their denominator has a possibility of the denominator being zero. According to the Unit Circle, secant and tangent are undefined at pi/2 or 3pi/2 when x=0. Cotangent and cosecant are undefined at 0 and pi when y=0.

BQ #2: Unit T Concept Intro


    BQ #2 
    How do the trig graphs relate to the Unit Circle?
    Kirch's SSS Packet

    a.) 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 when you go around the Unit Circle, you have to go around the entire circle in order for the patterns to repeat again. Sine's pattern in the quadrants is positive, positive, negative, negative and we see a pattern. Therefore the period would be 2pi because you had to go around the unit circle an entire time to see the repetition. For cosine the same applies but the pattern is positive, negative, negative positive. And you would also have to go around the unit circle entirely to see the pattern. So basically it takes 1 entire time around the Unit Circle to get repeating patterns for both sine and cosine which is 2 pi.

    The period for tangent is pi because you only need to go around the Unit Circle half way to get a repeating pattern. So it would be positive, negative, positive, negative to see a distinct pattern. Half of the unit circle is pi.

    b.) 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?
    The range of the Unit Circle values is from -1 to 1 so it determines the value of the amptitude. Sine's ratio is y/r and r=1 on the unit circle so you would only be limited to using one. Similarily, cosine's ratio is x/r, where r also equals 1, so you could only use 1 as well. Therefore, both sine and cosine have an amplitude of 1.

    Tuesday, April 1, 2014

    Reflection# 1: Unit Q: Verifying Trigonometric Identities

    1.  In Unit Q, we learned how to verify functions. When we verify functions, it basically means that we use the ratio identities, reciprocal identities, and the pythagorean identities in various order to simplify the equation further. We would only use the identities on the left side and NOT use the identities on the right side. We just want the right side to equal the newly simplified left side.

    2. To begin with, when verifying expressions make sure to list out the steps made neatly, so later on you can pinpoint mistakes. I found it useful to convert everything into sine and cosine. When we convert the expressions, then it would lead to canceling out terms and lead to a Pythagorean identity that equals zero. Also, you can factor out a least common denominator because it could lead to finding more properties within the expression.

    3. First thing I think about when I look at an identity is to see what identities are present in the problem.  Then I see if there is a greatest common factor to factor out to make it easier to simplify. I also look to see if I need to get a common denominator. If there is a binomial in the denominator then you would multiply by the conjugate. You could also combine or separate denominators.  TIP: Memorize the identities and know where you can use them, it will make it EASIER to solve.