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Coordinate geometry and trigonometry Introduction 3 Angles 4 Classifying angles 4 Relationship between angles 5 Parallel lines 8 Alternate angles 8 Corresponding angles 9 Co-interior angles 9 Triangles Types of triangles 12 12 Similar triangles 14 Quadrilaterals 17 Special quadrilaterals Pythagoras’ theorem Using Pythagoras’s theorem Sides and angles Adjacent and opposite sides The trigonometric ratios—sin, cos and tan 20 20 24 24 30 Labelling angles 35 Similar triangles—a reminder 35 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 17 1 2 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Introduction In this module we will mainly be learning about trigonometry and coordinate geometry. However, both of these topics will be referring back to various aspects of geometry and Pythagoras’ theorem. You have already done some work in both of these topics so we will just be going over those parts, which are relevant to this module. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 3 Angles Classifying angles An angle that is less than 90º is called an acute angle. A 90º angle is a right angle. An angle between 90º and 180º is called an obtuse angle. A 180º angle is a straight angle. An angle greater than 180º is called a reflex angle. A 360º angle is a revolution. Naming an angle When naming an angle make sure the middle letter is the vertex. This angle may be named as PQR or RQP. 4 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Activity 1 Classify and name each angle. Answers to Activity 1 1 CKQ or QKC is obtuse. 2 PTZ or ZTP is a straight angle. 3 GHL or LHG is an acute angle. 4 ACA is a revolution. (Note: Usually you are not asked to name revolutions.) 5 TNF or FNT is a right angle. 6 WSK or KSW is a reflex angle. (To distinguish a reflex angle from the smaller angle on the same diagram we usually write ref WSK.) Relationship between angles Complementary angles These are two angles that add to 90º. Supplementary angles These are two angles that add to 180º. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 5 Angles at a point The sum of all the angles at a point is one revolution or 360º. xº + yº + zº = 360º Vertically opposite angles When two lines intersect, two pairs of equal vertically opposite angles are formed. Activity 2 1 What is the complement of 36º? 2 What is the supplement of 58º? 6 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 3 Find the value of the pronumeral. Answers to Activity 2 1 54º 2 122º 3 (a) 62º (g) 54º (b) 127º (h) 52º (c) 324º (i) 317º (d) 25º (j) 235º (e) 21º (k) 328º (f) 96º (l) 37º 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 7 Parallel lines Alternate angles The angles marked with a bar are called alternate angles. Alternate angles formed on parallel lines are equal. There are two pairs of alternate angles. 8 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Corresponding angles The angles marked with a bar are called corresponding angles. Corresponding angles formed on parallel lines are equal. There are four pairs of corresponding angles. Co-interior angles The angles marked a and b are called co-interior angles. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 9 Co-interior angles on parallel lines add to 180º. They are supplementary. There are two pairs of co-interior angles. Activity 3 1 10 Name the types of angles shown in each diagram. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 2 Find the value of the pronumeral. Answers to Activity 3 1 2 (a) corresponding angles (f) alternate angles (b) alternate angles (g) corresponding angles (c) co-interior angles (h) co-interior angles (d) corresponding angles (i) (e) corresponding angles alternate angles (a) 106º (b) 74º (c) (d) (e) (f) 106º 112º 20º 180 – 50 = 130º 360 – 130 = 230º x = 230º (g) 320º (h) 30º (i) 140º 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 11 Triangles Angles in a triangle add to 180º. Types of triangles Scalene triangle All sides different length All angles different size Isosceles triangle Two sides equal Two angles equal Equilateral triangle 12 All sides equal All angles equal 60º 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Right-angled triangle Contains one right angle Activity 4 Find the size of the missing angle in 1–4 and then name the type of triangle. Name the type of triangle in 5–7. Answers to Activity 4 1 50º; isosceles triangle 2 90º; right-angled triangle 3 60º; equilateral triangle 4 65º; scalene triangle 5 Isosceles triangle 6 Equilateral triangle 7 Scalene triangle 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 13 Similar triangles Similar triangles are identical in shape but different in size. In similar triangles: the corresponding angles are equal the corresponding sides are in the same ratio. Note: When we are referring to similar triangles, ‘corresponding’ just means matching. Example Find x. The triangles are similar because all angles of one equal the corresponding angles of the other. The fractions formed by matching sides are equal. (Matching sides are opposite equal angles.) Solve this equation by multiplying both sides by 21. Use your calculator. x=7 14 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Activity 5 Find the value of the pronumeral. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 15 Answers to Activity 5 16 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Quadrilaterals A quadrilateral is a four-sided shape. The angles in a quadrilateral add to 360º. Special quadrilaterals Trapezium One pair of parallel sides Kite Two pairs of adjacent sides equal One pair of opposite angles equal Parallelogram Two pairs of parallel sides Two pairs of equal sides Opposite angle equal Diagonals bisect each other 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 17 Rhombus Has the same properties as a parallelogram Extra properties: All sides equal Diagonals perpendicular to each other Rectangle Same properties as a parallelogram Extra properties: All angles 90º Diagonals equal to each other Square Same properties as a rectangle Extra properties: 18 All sides equal Diagonals perpendicular to each other 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Activity 6 Name the quadrilaterals. Answers to Activity 6 1 Square 2 Parallelogram 3 Trapezium 4 Kite 5 Rectangle 6 Rhombus 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 19 Pythagoras’ theorem Earlier, you studied Pythagoras’ theorem. Since it will be needed when we do trigonometry, we will briefly revise it. In this right-angled triangle (ABC), a is the hypotenuse and b and c are the other two sides. For this triangle, Pythagoras’ theorem states: a2 = b2 + c2. Remember: 1 The side opposite the right angle is the hypotenuse. 2 The hypotenuse can be named in two ways, using the two capital letters at the ends of the line—BC—or using the one small letter ‘a’. Using Pythagoras’s theorem Pythagoras’ theorem is used when we know two sides of a right-angled triangle and want to find the third side. Example First it’s always a good idea to state the theorem: a2 = b2 + c2 20 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Now substitute the numbers and letters you know—make sure the hypotenuse is by itself. Example This is an equation. To get x2 by itself, subtract 92 from both sides. Notice how in the first example the numbers were next to each other so we added. In the second example the numbers are separated by the equals sign so when we moved them together we subtracted. Some students get confused when to add and when to subtract; if you set your work out as shown above you should have no problems. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 21 Activity 7 1 Find the length of the hypotenuse (correct to one decimal place). 2 Find the length of the short side (correct to two decimal places). 3 Find the length of the missing side (correct to one decimal place). 22 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Answers to Activity 7 Answers to one or two decimal places. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 23 Sides and angles Trigonometry has been used by people since the time of Alexander the Great (about 300 BC). In those early days, trigonometry had two main uses. The first of these involved practical applications in navigation and land surveying. The second was the rapid development of astronomy and the measurement of the earth, moon and sun. Nowadays, trigonometry is still used for investigating the universe and in navigation and surveying. It also has many other practical uses. Trigonometry is about the relationships between the sides and the angles of triangles. In this part of the section we are going to look at some of these relationships. Adjacent and opposite sides In the notes on Pythagoras’ theorem you found out that the side opposite the right angle is called the hypotenuse. In trigonometry the other two sides are also named. They are called the adjacent and opposite with reference to the two other angles. Here is the triangle ABC in which B = 90. On the diagram, write in the names of the sides: a, b and c. Does your triangle look like this? 24 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 b is the hypotenuse because it is opposite the right angle at B. Now look at the angle C and the other two sides of the triangle: The side c is opposite C. The remaining side a is adjacent to C (side between the marked angle and the right angle). So, when we are considering C: the side c is opposite C the side a is adjacent to C. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 25 Now look at ABC again, but this time consider A: Again b is the hypotenuse. The side a is opposite A. The remaining side c is adjacent to A. The hypotenuse is always opposite the right angle and is never called an adjacent side. Notice that: b is the hypotenuse a is opposite to A and is adjacent to C c is opposite to C and is adjacent to A. Look up the word ‘adjacent’ in your dictionary. Can you see why it’s used for the side lying alongside the angle? 26 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Activity 8 1 Here is the right-angled triangle PQR. Which of these statements are correct? (a) p is the hypotenuse (b) p is adjacent to R (c) r is opposite to R (d) r is adjacent to Q (e) p is opposite to Q (f) q is adjacent to R. Now fill in the words missing from these sentences: r is ______________________ to R and is ____________________ to Q. q is ______________________ to Q and is ____________________ to R. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 27 2 You are given enough information in this table to enable you to fill in the missing sides and angles. The first one has been done for you. Triangle Angle Hypotenuse Opposite side Adjacent side Y x y z B N s p Answers to Activity 8 1 The correct statements are: (a) p is the hypotenuse (c) r is opposite to R (d) r is adjacent to Q (f) q is adjacent to R. Did you fill in these missing words correctly? r is opposite to R and is adjacent to Q. q is opposite to Q and is adjacent to R. 28 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 2 Triangle Angle Hypotenuse Opposite side Adjacent side B c b a N l n m S t s r R q r p Did you do these correctly? Well done! If you did not, go over the notes on sides and angles again. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 29 The trigonometric ratios – sin, cos and tan Remember that the sides of a triangle are named with reference to a given angle, as shown here with reference to angle x. In trigonometry: We abbreviate sine to sin (still pronounced sine), cosine to cos and tangent to tan. 30 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 So in the triangle drawn below: sin x = = cos x = = tan x = = opposite hypotenuse O H adjacent hypotenuse A H opposite adjacent O A This rhyme may help you remember these ratios: SOH CAH TOA This reminds us that: sin x = O H (S is O over H) cos x = A H (C is A over H) tan x = O A (T is O over A) 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 31 Example In triangle DEF we will use angle F to determine the trig ratios. As shown below, in all trig questions you should label the sides as O, A and H before you start. In this triangle: sin F = O H cos F = A H tan F = O A = f e = d e = f d For the triangle below, complete the missing parts in the statements that follow: sin A = = tan C = = 32 O H b O A cos A = A H tan A = O A = c = a cos C = A H sin C = O H = = 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Does your work look like this? O H sin A = cos A = A H tan A = O A = a b = c b = a c tan C = O A cos C = A H sin C = O H = c a = a b = c b Well done! Now try these questions: 1 sin X = H = = cos Z = sin Z = H = 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 cos X = = A tan X = A = O tan Z = O = 33 2 sin P = cos Q = tan Q = = = = Do your answers look like these? 1 O H 4 = 5 cos X = A H 4 = 5 sin Z = O H 6 = 10 3 = 5 cos Q = sin X = cos Z = 2 sin P = A H 3 = 5 tan X = O A 4 = 3 O H 3 = 5 tan Z = A H 6 = 10 3 = 5 tan Q = O A 3 = 4 O A 8 = 6 4 = 3 If you had all of these answers, well done! If not, you should go back over the work on trigonometric ratios before going on. 34 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Labelling angles When we are using right-angled triangles in trigonometry, we often don’t label all of the vertices of the triangle. Pronumerals (such as x, y and z) or Greek letters are often used to indicate the angle we are considering. Commonly used Greek letters are (theta) and (phi). For example, in this triangle: tan = c b sin = c a cos = b a Similar triangles—a reminder Similar triangles are triangles which have the same shape but are different sizes. Similar triangles are equiangular and their sides are in proportion. Equiangular means that the three angles of one triangle are equal to the three angles of the other triangle (or triangles). 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 35 The diagram below shows three similar right-angled triangles. For Triangle 1, the side opposite to A is 12 units and the side adjacent to A is 10 units. O A 12 = 10 So the value of tan A = = 1.2 For Triangle 2, the side opposite to A is 6 units and the side adjacent to A is 5 units. O A 6 = 5 So the value of tan A = = 1.2 For Triangle 3, what is the value of tan A? Did you find that the value is also 1.2? So we have discovered that tan A = 1.2 in all three triangles, irrespective of the size of the triangle in which it is found. 36 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Check that the values of both sin A and cos A are the same in all three triangles. You will have to use Pythagoras’ theorem to find the length of the hypotenuse in each case and then use your calculator to get a decimal value so that you can compare your answers. Did you get these results? sin A = 0.77 (to two decimal places) cos A = 0.64 (to two decimal places). Well done! We have now discovered that sin A, cos A and tan A have the same values in similar triangles, irrespective of the size of the triangle. Activity 9 1 Here is the right-angled triangle JKL. Select the correct statements from this list: (a) sin L = l k (b) tan K = k l (c) cos L = j k (d) cos K = l j (e) tan L = l k 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 37 2 For this right-angled triangle, write down the tangent, sine and cosine ratios of . 3 Complete these statements for triangle LMN above: tan N =…. sin M =…. cos N =…. 38 cos… = n l tan… = m n sin… = n l 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Answers to Activity 9 1 The correct statements are: k l l (d) cos K = j (b) tan K = (e) tan L = 2 3 tan = l k y z y sin = x z cos = x n tan N = m m sin M = l m cos N = l n cos M = l m tan M = n n sin N = l Did you notice that sin M = cos N for this triangle? Congratulations! You have completed Section 1 of this module. Have a break before starting the Check your progress exercises. These will revise all of the work covered in this section. If you have any difficulties, go back and revise the appropriate examples. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 39 Check your progress 1 What is the complement of 72º? 2 What is the supplement of 58º? 3 Find the value of x. 4 Name the types of triangles. 5 Find the value of x. 40 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 6 Name the quadrilaterals. 7 (a) Find x correct to two decimal places. (b) In ABC below, C = 90º, c = 40 cm and a = 24 cm. Find the length of AC. (c) In the triangle below, find the value of h correct to one decimal place. 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 41 8 Complete the following table referring to UVW. Angle Hypotenuse Adjacent side Opposite side V W 9 Here is a right-angled triangle RST. Complete the following statements: 42 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 Answers to Check your progress 1 90º – 72º = 18º 2 180º – 58º = 122º 3 (a) x = 90º – 21º = 69º (b) (c) (d) (e) x = 180º – 53º = 127º x = 360º – 158º = 202º x = 27º (vertically opposite angles) x = 180º – 118º = 62º (co-interior angles) (f) x = 118º (corresponding) (g) x = 118º (alternate). 4 (a) equilateral triangle (b) right-angled triangle (c) isosceles triangle (d) scalene triangle. 6 (a) square (b) (c) (d) (e) kite rectangle trapezium parallelogram (f) rhombus. 7 (a) 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303 43 (b) (c) 8 Angle Hypotenuse Adjacent side Opposite side V u w v W u v w How did you go? Hope you did well! You have now completed the first section in the module Coordinate Geometry and Trigonometry. 44 4930CQ: 1 Coordinate Geometry and Trigonometry 2004/045/020/11/2004 P0027303