The difference quotient is derived from the slope of a secant line. The difference quotient is of course f of x plus h minus f of x divided by the letter h. We know that the difference quotient is derived from the slope of a secant line because we would have to find the different quotient in order to then find the slope of the secant line. To find the derivative of the difference quotient we do all of this work but with the left over h's we have to plug in zero for them.
Once we do this then we set our equation equal to zero, eventually finding our slope intercept form.
Continuity is when there are no jumps, breaks, or holes. You can draw the line without having to raise your pencil. The limit and value have to be the same. A discontinuity is when there is a jump, break, or hole. There are different types of them jump, point, infinite, and oscillating.
2. What is a limit?
A limit is the intended height of the function.
When does the limit exist?
The limit exists when you reach the same height from the left and the right.
When does the limit not exist?
The limit does not exist when you don't reach the same intended height from both the left and the right or when you have a jump discontinuity and have the closed circle above the open circle.
What is the difference between a limit and a value?
The difference is that a limit is the intended height of the function the value is the actual height of the function.
3. How do we evaluate limits numerically, graphically, and algebraically?
Numerically is when we use a table chart to help us evaluate the limit we pick a number and get closer to it from the left and the right. Graphically is when we use a graph as a visual and use our fingers to see if the limit has the same left and right. Algebraically is when we use substitution, division, factoring,and rationalizing to help us evaluate the limit.
Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?
The difference between these two graphs is their asymptotes, when you plot the sine graphs you locate the asymptotes by looking where the graph touches the x-axis. When you do locate the asymptotes then you will see that for tangent and cotangent they are different. One will start positive and end negative whereas the other will start negative and end positive. We know that the order of the quadrants go positive,negative,postive,negative and when you find the asymptotes both graphs will be different because they either shift right or left which changes how they are placed and drawn on the graph.
How do the graphs of sine and cosine relate to each of the others?
Remember that sine and cosine have asymptotes because they are undefined when a zero is placed as a denominator, you can't divide by zero. To relate these to all the other trig functions we know that sine and cosine are involved with their ratios.
For tangent we know that it is equal to sine over cosine. We can see that tangent will have asymptotes at 3pi/2 and pi/2.
Cotangent is the exact opposite of tangent instead of being sine/cosine it is cosine/sine.
Secant will always be like the graph for cosine but the only thing that is different is the asymptotes, secant is related to cosine because it is the the reciprocal 1/cos is equal to secant.
Cosecant follows the graph of sine and it is the reciprocal of sine,1/sin equals Cosecant. The difference again is the asymptotes.
Why do sine and cosine not have asymptotes, but the other four trig functions do?
Sine has a ratio of opposite over hypotenuse and cosine has a ratio of adjacent over hypotenuse. In the unit circle the hypotenuse is always going to be one. You can't have any asymptotes because in order to get asymptotes on the graph you would need a zero to be divided and you can't divide by zero if you do zero is undefined. When a trig function is undefined it means it has asymptotes and since sine and cosine are not undefined they do not have asymptotes.
The period for sine and cosine is 2pi because that's how much the distance is for the pattern of sine and cosine to repeat they have to go a full revolution in the unit circle whereas the period of cotangent and tangent is pi because it only takes a half a revolution for these two to repeat their patterns.
The unit circle only reaches a radius of one, we all know that, and sine and cosine have one as their amplitude because of the unit circles radius. Other trig functions don't have restrictions which is why they don't have amplitudes
What does it actually mean to verify a trig function?
To verify a trig function is when you have one of the problems like in one of the concepts 1 or 5 and then just equaling the left side to the right side. It is like you are given a puzzle and you have a picture of the puzzle completed but you only have puzzle pieces. You have to put the pieces together to get it to look like the picture. It's as simple as that, but contains more math and is more challenging.
What tips and tricks have you found helpful?
Some tips that I recommend are to look at the problem and look for simple things to do or change don't over think the problem because the answer will fly by your head. Some tricks that I use is to actually memorize the identities that way you don't have to keep referring to your SSS packet.
Explain your thought process and steps you take in verifying a trig identity.
My thought process first is to recognize what I have and look for simple things to do. I always try a few things like looking for a GCF, Substituting an identity, The conjugate, CLT, Separating fractions, and factoring. I try looking to see if any of these will work most of the time the problems are simple, but I overthink everything and do way more than I have to. Then make sure to write the steps that you have taken in order to receive credit, but to also help you remember what you had done in order to get that answer. Personally I think verifying is easier than simplifying because verifying already gives you the answer and when you simplify you are not sure whether you are right or not.
In earlier units we learned about the three basic trig functions which are sin, cosine, and tangent. Sine which means x/r so the y is the side that rises and r is the hypotenuse. R replaces the c^2 in the Pythagorean theorem. So it would really be a^2+b^2=R^2. Cosine we know is y/r . We know that when looking at the problem above that it really is x/r^2+y/r^2=1. Using the unit circle to help prove that this statement is true is fairly easy we use the 45-45-90 triangle. Which is square root of 2/2, square root of 2/2. When squaring these you get one half plus one half equals one. Which in this case is an identity which means it is a true proven statement.
2. Show and explain how to derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1
To derive the two remaining identities all you have to do is divide by either cosine or sine. We know that if we divide by the same function we will get one so that is how we get the one on the right side. We will also notice that if we divide by a trig function like sin/cos we simplify that to x/r /y/r we multiply the reciprocal of the denominator to top and bottom to get the new one which is tangent. You do the same thing for the other identity.
Inquiry Reflection Activity
1. The connection I see between Units N, O, and P are that we still have to use the unit circle in some we and that we still use the trig functions even more.
2. If I had to describe trigonometry in 3 words, they would be challenging, depressing, frustrating.
Please see my WPP13-14, made in collaboration with Ivan L, by visiting their blog here. Also be sure to check out the other awesome posts on their blog
1. Law of Sines We need the law of sines to help determine the missing sides of an oblique triangle. We can also use it to determine the sides of a right triangle. Sin is opposite/ hypotenuse. When we have a problem we will always be given an angle we use one of those angles to help us determine one of the missing sides. The law of sines comes from trig. Sine is only used for right triangles but with non right triangles we can use them too. When we use some we have to make sure the question gives us AAS or ASA 4. Area Formulas
Always remember that the area of a right triangle will always be 1/2 bh. B is the base and h is the height. We will not always be given the height so we have to find the height using sine. You use the same method of finding the area of a right triangle for finding the one of an oblique triangle. You split the oblique triangle in half or wherever you can so you can determine the height. Use an angle given to you to determine one of the sides. You keep going until you find the height and use the same equation. It will always relate to the area of a triangle formula because we have to use it in order to get what we are looking for.
Problem: Julian decided to take a boat ride on his brand new yacht. He takes it out for a spin and notices a cliff upcoming ahead. He wonders, if the angle of elevation to the top of the cliff is 52.53 degrees, and the base of the cliff is 1236 feet away from the boat, how high is the cliff? (Remember round to the nearest foot)
Ivan was on top of a lighthouse observing Julian on his brand new yacht. He looked around and noticed at the bottom of the cliff where he was positioned, that there was a boiling crab. He found a parachute and thought, " I wonder how long will be the path I glide if the angle of depression is thirty degrees?" Keep in mind he is already 300 feet above ground.
For this activity we had to derive the special right triangles not from the unit circle though. We had to derive the two special kinds ,45-45-90 and 30-60-90 right triangles. Each are completely different we had to find out what n was and why n couldn't just be a number.
How can we derive the 45-45-90 triangle from an square with a side length of 1?
We know that if we are given a square with side lengths of 1 we will have two right triangles if we split the square in half from the corners. For a 45-45-90 triangle we know the base and height are the same,1. If we do the Pythagorean theorem we will see that the hypotenuse will equal radical 2. N is there to be any given value, without n the triangle sides cannot be altered to match the initial constants, it is multiplied by the initial constants. For example if n were to equal 2 then the base and height are 2. The hypotenuse will equal 2 radical 2.
How can we derive the 30-60-90 triangle from an equilateral triangle with a side length of 1?
To derive the pattern for the 30-60-90 triangles we have to cut an equilateral triangle in half straight down the middle. Each side length of the triangle is one so when we split the triangle in half the base turns into 1/2. We also split the triangle in half to get 30 degrees as one of the angles. Then we will have a 30-60-90 triangle. Then we notice that we do not have the height of the triangle, we have to use the Pythagorean Theorem. Once you get the constants they can be altered to get rid of the ugly fractions by multiplying by two ( look at picture). The constants are the same because they are all proportional and everything was multiplied equally.
Inquiry Activity Reflection:
Something I never noticed before about special right triangles is how we have to tweak equilateral triangles and squares to get the sides for n.
Being able to derive these patterns myself aids in my learning because now I can refer to this square or triangle if I ever forget what the sides of the triangle equal.
The 30* triangle has three different sides to it: adjacent which is x, opposite which is y, and hypotenuse which is r. The side opposite the hypotenuse will always be x.The hypotenuse must be one if you want to derive the unit circle from the triangle. If you want this to happen you have to divide each side by 2x. Once you do this you will get x=radical 3 divided by 2, y=1/2 and r=1. We can use these simplified values as coordinates to determine where 30 degrees lies in a quadrant on the unit circle. We know that 30 degrees on the unit circle is located on radical 3/2, 1/2. This can be used for 150 degrees, 210 degrees, and 330 degrees. The only difference is that they are located on different quadrants and there will be negatives and positives.
2. The 45* Triangle
The 45* triangle has two sides that are the same length which are x and y the hypotenuse is r. To derive the unit circle from the triangle we have to get the hypotenuse to equal 1. In order to do this we divide every side by x radical 2. Once we get one on the hypotenuse we can get r and figure out the points. For the angle of 45 degrees we plot radical two over two, radical two over two. This is where it will lie on the unit circle. This will also stand for 135, 225 and 315 degrees. The only differences are the quadrants the negatives.
3. The 60* Triangle
The 60 degree triangle is the same as the 30 degree triangle it has three different sides. In order to derive this triangle from the unit circle is to divide the hypotenuse to get one. This is the same as the 30 degree triangle because you divide by 2x. When you get your final answers you will be able to plot the points. The points for 60 degrees are 1/2, and radical 3/2. These rules also apply for 120,240, 300 degrees.
4.
This activity helps me derive the unit circle because the triangles reflect different points on the unit circle throughout all of the four quadrants. Each of these triangles can be found in all of these quadrants and are all the same the only exception is that there are negatives and positives and they are located in different quadrants too.
5.
The triangle in this activity lies in quadrant one both the x and y values are positive which means that it is in quadrant one.
Inquiry Activity Reflection: 1. The coolest thing I learned from this activity was how you can find the points on the unit circle by using the special triangles. 2. This activity will help me in this unit because it can help me memorize where different points are and where some points lie on different quadrants. 3.Something I have never realized before about the special right triangles and the unit circle are that both these are in relation to each other when the hypotenuse is equal to one
All there is you need to know about Hyperbolas 1. Mathematical Definition. A hyperbola is "a curve where the distances of any point from a fixed point 9 (the focus)and a fixed straight line (the directrix) are always in the same ratio."(mathisfun.com)
These two pictures show the two formulas used for a hyperbola. The reason there are two different formulas is because the way each hyperbola branches out. This means that if the equation begins with x it will open up on the x- axis. You will also know that it is the horizontal transverse axis. If the equation begins with y it will open on the y-axis and you will also know that it is the vertical transverse axis. The different parts of this conic section on this axis are vertices, foci, and center. The conjugate axis includes co-vertices. The conjugate axis is the direction in which the hyperbola opens. Asymptotes are also very important they are shown as y=mx+b. The formula to find the asymptotes for a transeverse axis which is horizontal us y=k+ or - b/a (x-h). The formula to find the asymptotes for a transverse axis which is vertical is y=k + or - a/b (x-h).
To find the center which is (h,k) they will be included in the formula so they are paired off with x and y. X is always with h and y is always with k. To find the vertices you use what you have already which is the center. For example if the hyperbola y is first then you find the center and then use the x from the center to find the vertices which is the number for a up and the number for a down. The same with the co-vertices except you go left and right. A and B are found when you take the square root of the denominators. If y is first a squared is underneath it and b squared is underneath x or vice versa. To find the asymptotes you use the formula given in the above paragraph.
This video explained how an ellipse and a hyperbola are similar even with their foci. The eccentricity of a hyperbola should be 1 or above.The closer the eccentricity is to one the curves of the hyperbola will make it look sharper or more pointy. The farther away the eccentricity is to one it will make the curves of the hyperbola look straighter, which means it will look fatter. A smaller foci means a small width and a larger foci means a larger width.
Hyperbolas can be found all over the real world for example a common household lamp can cast a shadow that is a hyperbola. Also in architecture like the Dulles Airport which is shaped like a hyperbola. The one i picked is a nuclear power plant's cooling tower. The reason that this is the standard shape for all cooling towers is because it needs to withstand high winds and be built with little material. When the cooling tower is in the shape of a hyperbola it is easier for the building to take the damage of high winds. This shape also is very inexpensive to build with steel beams and concrete.
As you can see the cooling tower looks like a cylinder. Well when you take a cylinder by each side and push one side forward and one side back you will create a hyperbola. Hyperbolas can also be found with jets. As a jet breaks the sound barrier it releases a sonic boom. When the boom is released a cloud is formed and it looks like a cone, when it comes into contact with the ground then it becomes a hyperbola.
4. References
Vertical and Horizontal Transverse Axis of Hyperbola: http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php http://www.mathwarehouse.com/hyperbola/graph-equation-of-a-hyperbola.php
