Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
STRAND 2 REVIEW Working through these questions will help you assess your understanding of the learning outcomes listed here: Question type Review Level All Learning outcome use of the theorem of Pythagoras to solve problems (2D only) use the sine and cosine rules to solve problems solve problems that involve calculating the cosine, sine and tangent of angles between 0˚ and 90˚ graph trigonometric functions of type aSin n , aCos n for a, n N 1 Q. The diagram represents a plan for a new playground slide B 5m D 1m C 0.5m 2.5m m E F G A Calculate the height of the slide. Health and safety will not allow a slide in the playground if the angle between the ground and the slide is bigger than 27˚. Does the planned slide comply with safety regulations? Look at the sketch of the slide. Can you see where the steps would be? Where the bit you slide down would be? Can you identify the ground? Where is the angle that must not be greater than 27 ? The angle between the slide and the ground How many triangles can you see in the diagram? Can any of these triangles help you to find the height of the slide or the size of the angle that must not be bigger than 27 ? Can you use Pythagoras Theorem? Can you use the trigonometric ratios Sin , Cos or Tan ? Will the slide comply with Health and Safety regulations? Make a decision based on your evidence 2 Q. Can a triangle have more than 2 obtuse angles? Explain. What is an obtuse angle? If you add two obtuse angles together what is the smallest number you can get? When you add all the angles in a triangle together what must they add to ? Q. John is painting the house. He needs to use a stick to stir the paint. He doesn’t want to get paint on his fingers so wants to make sure the stick is long enough A diagonal line drawn on the diagram of the paint tin represents to smallest stick that could be used. Draw this line. Using a scale of 1cm=4cm Take measurements from the diagram and use these to calculate the minimum length the stick can be. Draw the diagonal line. Can you see a triangle? What is the height of this triangle? The base? Is it right angled? Try using both Pythagoras Theorem and the trigonometric ratios Sin , Cos or Tan to help find the length of the stick Do you get the same answer? 3 Q. The diagram below shows two submarines, B and C and their distances from a sonar station marked A. Point D represents an obstacle 2km from B, and 5km from A. How far is submarine C from the obstacle? 27 km 6 km 5 km 2 km Look at the diagram mark the distance you are being asked to find as x How many triangles can you see? Use the flow chart from the presentation found at https://emea67395290.adobeconnect.com/_a858841383/maths/ to help decide on a solution strategy Don’t forget to check if your synthetic geometry theorems or axioms can help 4 MIND MAPPING CAN YOU SEE A NO TRIANGLE? WILL MY SYNTHETIC GEOMETRY THEOREMS OR AXIOMS HELP? YES NO IS IT RIGHT CAN I USE THE ANGLED? SINE RULE? YES NO YES CAN I USE PYTHAGORAS’ NO YES WHAT WILL THIS HELP ME FIND? WHAT WILL CAN I USE THE THIS HELP COSINE RULE? CAN I USE THE YES TRIGONOMETRIC RATIOS – SINE, COSINE, TAN? WHAT WILL THIS HELP ME FIND? YES WHAT WILL THIS HELP ME FIND? 5 Q. Use the notes on the next page to helpyou with this. (a) Show that sin (x + 2π ) = sin x (b) Draw the graph of the function y = sin x where x (c) R in the domain -2π ≤ x ≤ 2π On the same axis sketch the graph of the function y = Sin 3x. What is the period of this function? 6 FACT When the values of a function regularly repeat themselves, we say that the function is periodic. By this definition is the function sin x periodic? Back up your decision with a table graph DEFINITION If, for all values of x, the value of a function at (x + p) is equal to the value at x. That is: If f(x + p) = f(x) then we say that the function is periodic and has period p. What is the period of sin (x+2 )? Back up your decision with a table a graph an equation [Try to write sin (x + 2π) in a different way? This will help you to show that sin (x + 2π) = sin x. Look in the tables for help on how to write sin (x + 2π) in a different way] Think about the following: sin1x, sin 2x, sin 3x…….sin ax. What does the a stand for? Use Geogebra to draw these graphs. What is different about each one? Focus on the x axis in the interval between 0 and 2π .How many times does the pattern repeat itself in this interval? 7 What does the a stand for? Number of times pattern repeats itself in the 0 -2π interval Sin1x Sin2x Sin3x Sin ax Conclude What is the period of the function Sin ax? Think: Can you decide on the period of sinax depend on the period of sinx? 8 Question type Review: Strand 2 Level LC /HL Learning outcome This question provides you with an opportunity to display evidence that you can calculate the area of a triangle solve problems involving the length of the perpendicular from a point to a line recognise that x2+y2 +2gx+2fy+c = 0 represents the relationship between the x and y co-ordinates of points on a circle with centre (-g,-f) and radius r where r = √ (g2+f2 – c) recall that a diagonal divides a parallelogram into two congruent triangles. On the grid provided plot the points A (0, 4) B (4, 0) and C (-2,-2). 9 a) (i) Find the mid-points of AB, BC and AC label these points X, Y and Z respectively Join X to Y, Y to Z and X to Z. (ii) What can you say about the area of ∆XYZ in terms of the area of ∆ABC? Show that this is true b) Find the equation of the circumcircle of ∆ABC. Note to Students: There are several ways you may approach this question. a) In order to find the midpoint you may use the co-ordinate geometry formula to find the co-ordinates of the midpoints or you could simply measure the length and half it. Which method do you think will give you the most accurate result? Why is this? You should note that the area of ∆XYZ is ¼ that of ∆ABC. Why is this true? The truth of this statement can be shown in various ways. You may calculate the area of each triangle, or you can show that AXYZ is a parallelogram which means that ∆AZX and ∆XZY are congruent triangles. Similarly ZXBY is a parallelogram making ∆XZY and ∆XYB congruent pairs. A similar argument means that ∆CZX is congruent to ∆YXB. So ∆ ABC is made up of 4 congruent triangles meaning that ∆ XYZ is ¼ the area of ∆ABC b) The equation of the circumcircle can be found by substituting the vertices into the general equation of the circle, or you could construct the circumcentre and calculate the length of the radius and substitute these values into the equation. Try all of the methods mentioned in the note Think about other ways the question could be asked. Write some notes to help organise your thoughts about circles, triangles and circumcircles and how they are related. Follow the link http://www.mathopenref.com/trianglecircumcircle.html to investigate properties of circumcircles of triangles. When is the centre of the circle inside the triangle? outside the triangle? Visit http://www.mathopenref.com/constcircumcircle.html here you will get help on how to construct the circumcircle of a triangle. Think about the mathematics of each step why will these steps result in you being able to construct the circumcircle of a triangle? 10 11