Pythagoras' theorem and trigonometry (2)

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CHAPTER 31

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Pythagoras’ theorem and trigonometry (2)

CHAPTER

Specification reference: Ma3.2g

Chapter overview       

Understand, recall and use Pythagoras’ theorem in solving problems in 3 dimensions Understand, recall and use trigonometry in solving problems in 3 dimensions Identify and work out the size of the angle between a line and a plane Recognise, describe, draw sketches of, and use graphs of y  sin x°, y  cos x°, y  tan x° Solve simple trigonometric equations Understand and use 12 ab sin C to calculate the area of a triangle Understand and use the sine and cosine rules in triangles and in solving problems in 2 and 3 dimensions

31.1 Problems in three dimensions H Demand Grade

In this sections … L DC

M B

H AA*

By the end of this section students will:  Be able to apply and use Pythagoras’ theorem and trigonometry in 3 dimensions.  Be able to identify and work out the angle between a line and a plane.

Prior knowledge Students should already: Be able to use Pythagoras’ theorem and trigonometry in 2 and 3 dimensions. Know the basic properties of cuboids, cubes, pyramids and prisms. Know and understand the terms ‘vertex’, ‘edge’ and ‘face’ as they apply to a 3-dimensional shape. Checking prior knowledge Non-ICT Starter 31.1 Copy this triangle, onto the board. Ask a student to tell you Pythagoras’ theorem for this triangle. (c2  a2  b2) Label one of the angles ‘x’ and as a class, write down the sine, cosine and tangent of angle x in terms of a, b and c. b a b sin x   cos x   tan x   c c a

b

c x a

What is a prism? (3-D shape with a cross-section that is the same all along its length, the cross-section is constant) How many faces has a cuboid? (6) How many vertices has a square-based pyramid? (5) How many edges has a triangular prism? (9) TEACHING THE TOPIC

Key vocabulary and phrases Cuboid Cube Pyramid

Prism

Three dimensions

Angle between a line and a plane

Non-ICT Main Activity 31.1 Draw this cuboid on the board. As a class, find the 2 a) length BD ((12   72)  193   13.9 cm) 2   102  193   100  293   17.1 cm) b) length BH (BD 10 c) size of angle FDB (tan FDB    0.7198, angle FDB  35.7º)  193 268

H

G

E

F D

10 cm

C 7 cm

A

12 cm

B

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31.1 Problems in three dimensions

Non-ICT Main Activity 31.1 (contd) Explain that a table-top is a horizontal plane. Hold a large ruler so that one end rests on the top of the table and incline the ruler so that it makes an acute angle with the table-top. Estimate the size of the angle between the line (ruler) and the plane (table-top). With the help of a student and a piece of thread attached to the top of the ruler, show that many angles can be formed. And indicate how a right-angled triangle can be formed using the ruler, table-top and an imaginary vertical line from the ruler to the table-top, at right angles to the table-top.) What information would we need to be able to find the size of the angle? (Length of two of the sides in the rightangled triangle.) TEACHER’S TIPS

 As long as students can solve problems using Pythagoras’ theorem and trigonometry in two dimensions with confidence, then by identifying and drawing suitable right-angled triangles from three-dimensional shapes, they should be able to approach problems in three dimensions with confidence. The key is to draw as many separate right-angled triangles as necessary.  Remind students to check that their calculator is in degree mode whenever they use trigonometry.  Remind students that, unless a question says otherwise, lengths should be given to 3 significant figures and angles to 0.1°. However, when a value from one part of a question is to be used in another part, the non-rounded value should be used.  A pyramid in which the vertex is vertically above the centre of the base when the base is standing on a horizontal plane, is called a right pyramid. However, this term is not used on GCSE papers so it has not been used here. Students should be familiar with the properties of such pyramids, such as the fact that the sloping edges will be equal in length.  Encourage students not to use the word side when they mean face.  There are many angles between a line and a plane, as demonstrated in the Main Activity. The angle between a line and a plane needs to be defined and the angle is taken to be the smallest angle between the line and the plane.  The correct mathematical term for the line AN in the diagram at the bottom of page 499 is ‘the projection of AB on the plane’.  Students do not need to know how to find the angle between two planes and the angle between two skew lines.  Remind students of the importance of drawing separately all relevant right-angled triangles and mark known sides and known angles with their sizes.

Individual activity: Exercise 31A This exercise has three questions, one on a cuboid, one on a triangular prism and one on a square-based pyramid. In each question students have to calculate lengths and calculate the size of named angles. Individual activity: Exercise 31B Q1 to 5 give basic practice at finding the angle between a line and a plane. Q6 gives the angles between lines and planes and students have to calculate lengths. Q7 requires students to understand how a pyramid with a square base and a cube are put together to make a solid. TOPIC SUMMARY

Key questions 1 Draw this cone on the board. The base of the cone is horizontal. The line AB is a diameter of the base and the vertex, V, of the cone is vertically above the centre of the base. The lines VA and VB are called slant heights of the cone. (a) Work out, correct to 3 significant figures, the length of VB. (17.2 cm) (b) Work out the size of the angle marked a in the diagram and hence work out, correct to the nearest degree, the size of angle AVB. (a  35.5376…°, AVB  71°)

V a 14 cm

A

10 cm

B

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2 Here is a cuboid. Name the angle between these lines and planes. (a) HB and the plane ABCD (angle HBD) (b) ED and the plane ABCD (angle EDA) (c) EC and the plane BCGF (angle ECF) (d) What is the size of the angle between HG and the plane ADHE? (90°)

H

G

E

F D

C

A

3 The diagram shows a triangular prism. The right-angled triangles ADE and BCF are congruent. ABCD is a horizontal rectangle. CDEF is a vertical rectangle. ABFE is a rectangle. (a) Name and find the size of the angle between EA and the plane ABCD. (Angle EAD, 45°) (b) Work out the length of FA. (41.2 cm) (c) Name and find the size of the angle between FA and the plane ABCD. (Angle FAC, 29°) (d) Name and find the size of the angle between FA and the plane DCFE. (Angle DFA, 29°)

B E

F 20 cm

C

D

20 cm

A

4* The diagram shows a triangular-based pyramid. This is called a tetrahedron. The plane ABC is the base of the pyramid and M is the midpoint of BC. The point N is on AM and is the foot of the perpendicular from the vertex D of the pyramid onto the plane ABC. Name the angle that is the angle between (a) DA and the plane ABC (b) DM and the plane ABC (c) DC and the plane ABC ((a) angle DAN (b) angle DMN (c) angle DCN.) 5* The diagram shows a solid cube of side 12 cm. The points P, Q and R are the midpoints of the edges on which they lie. The pyramid OPQR is removed from the cube. (a) Taking OPQ as the base of the pyramid, draw a sketch of the pyramid, marking the size of angles POQ, QOR and ROP and the lengths of sides OP, OQ and OR. (b) What is the size of the angle between (i) RP and the plane OPQ (ii) RQ and the plane OPQ? ((i) 45° (ii) 45°) (c) What is the volume of the solid formed when the pyramid is removed from the cube? (Volume of cube  1728 cm3, volume of pyramid  36 cm3, volume of solid  1692 cm3)

B

30 cm

D

C M N

A

B

Q R

O P R

12 cm

6 cm O 6 cm

Q

6 cm P

31.2 Trigonometric ratios for any angle H Demand Grade

In this section … L DC

M B

H AA*

By the end of this section students will:  Understand how to find the trigonometric ratios for any angle.  Recognise, describe, draw sketches of, and use graphs of y  sin x°, y  cos x°, y  tan x°.  Solve simple trigonometric equations.

Prior knowledge Students should already: Know how the sine, cosine and tangent of an acute angle can be found using a right-angled triangle. Understand coordinates in all four quadrants. Understand that there are 360° in a complete turn.

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31.2 Trigonometric ratios for any angle

Checking prior knowledge Non-ICT Starter 31.2

ICT Starter 31.2

Draw a right-angled triangle on the board with the perpendicular sides not horizontal or vertical. Label the smallest angle in the triangle . As a class, write down the equations for finding the sine, cosine and tangent of the angle . How many degrees are there in a complete turn? (360º)

Display ICT Starter 31.2 which shows a series of triangles and trigonometric questions. Students have 10 seconds to find the missing measurement from the triangle before the answer appears. Click ‘Play’ to start, and ‘Pause’ to stop. Use the arrows to adjust the time. Use ‘Back’ and ‘Next’ to move back and forth through the questions. ‘Restart’ returns the program to the start. Questions refer to diagrams, and involve the identification of the appropriate triangle to find missing values. Calculators are not required. Q15 involves knowing that cos 50o and sin 40o are equal, which can be deduced from previous questions.

TEACHING THE TOPIC

Key vocabulary and phrases sin ° tan °    (theta) cos ° Non-ICT Main Activity 31.2

ICT Main Activity 31.2

Draw a set of axes from x  1.2 to x  1.2 marking every 0.2 division on the board. Then ask students to imagine a rod (OP) of length 1 held fixed at the origin and rotating from the positive x-axis in an anticlockwise direction.

Display ICT Main Activity 31.2 which is a simulation showing a circle of radius 1 on a set of axes, with a line, OP, which can be rotated anticlockwise about the point O. The coordinates of the point P can be shown, and graphs drawn for sin °, cos ° and tan ° in the range 0o360o by selecting the appropriate function and rotating the line OP. A completed graph for sin ° and cos  ° can be displayed on the graph for comparison purposes, and the circle axes can be rotated 90o anticlockwise to help with the explanation of cos °.

What happens to the angle, , between the line OP and the positive x-axis as OP moves round the axes? (increases from 0° to 360°) P

y θ

Move point P in an anticlockwise direction around the circumference of the circle.

x

O

Draw a table of values from   180° to   540° at intervals of 45 as started below.

 sin  cos  tan 

180º 135º

(0)

(0.707)

(1) (0.707) (0)

(1)

90º (1)

45º

0º

(0.707) (0)

(0)

(0.707)

undefined

(1)

(1)

(0) etc.

As a class use calculators to find sin , cos  and tan  for each value. Ask students to work with a partner to plot the graphs of y  sin , y  cos  and y  tan  noting key features such as maximum and minimum points and points of intersection with the x-axis. continued 

What are the coordinates of P? Click ‘Show Coordinates of P’ to display the coordinates of P. What is being shown by the graph? What is being shown on the y-axis of the graph? How does this relate to the coordinates of P? What is being shown on the x-axis of the graph? What is sine of the angle shown? Check using a calculator. What do you notice about the form of the graph? Discuss maxima, minima and the repeating pattern of the graph. Click ‘Reset’ to return to the starting conditions, or drag the arm clockwise back to the start. Using the dropdown menu, select cos °. If required, rotate the axes on the circle using the ‘Switch Circle Axes’ button. To return to normal axes setting, click ‘Switch Circle Axes’ again. continued 

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Non-ICT Main Activity 31.2 (contd)

ICT Main Activity 31.2 (contd)

Solve 3 sin   2

Repeat as above for cos °. Once cos ° is drawn, Click ‘Show sin  °’ to display the graph of sin ° overlaid on the graph of cos °.

Discuss that while a calculator gives one solution (sin1   23,   41.8°) the graph of y  sin  can be used to show that there are other solutions. A range; e.g. 180°    180° is usually given in the question.

How does cos ° compare to sin °? Using the dropdown menu, select tan  and repeat as above. Note that as  approaches 90°, the x-coordinate y-coordinate becomes very small, so  becomes very x-coordinate large. At 90° the x-coordinate becomes 0, and the graph is discontinuous because it is not possible to divide by zero.

TEACHER’S TIPS

 In order to be able to draw, sketch and describe the graphs of trigonometric functions for angles of any size, students need to know that trigonometric ratios can be found for angles of any size. How much emphasis to put on the discussion of the rotating line OP depends on the ability of the students in a group. You will need to make this decision, bearing in mind that the important objective of this section is to be able to sketch and use the graphs of y  sin °, y  cos °, y  tan °. Students may be convinced by a consideration of the values from a calculator.  The graphs obtained by transformations of y  sin °, y  cos ° are considered in Chapter 36. Consequently only equations of the form a sin °  b, a cos °  b, a tan °  b are considered in this section.  The important properties of each graph are given on p 505 of the Student Book because the specification says that students should be able to describe the graphs of trigonometric functions.  In solving a trigonometric equation, students should sketch the graph of the relevant trigonometric function for the values of the angle given in the question.

Individual activity: Exercise 31C Q1 checks that students know how to sketch the graphs of y  sin °, y  cos °, y  tan °. Q2 involves equations of the form sin °  a, cos °  a, tan °  a. Questions 3 to 5 involve equations of the form a sin °  b, a cos °  b, a tan °  b, with some help being offered in Q3 and 4. TOPIC SUMMARY

Key questions 1 (a) sin 30°  0.5, what is sin 150°? ( sin 30°  0.5) (b) cos 60°  0.5, what is cos 480°? ( cos 120°  cos 60°  0.5) (c) tan 45°  1, what is tan 315°? ( tan 45°  1) 2 Solve each of these equations for 0  x  360. (a) sin x°  0.7 (44.4, 135.6) (b) 5 cos x°  2 (66.4, 293.6) (c) 3 tan x°  7 (66.8, 246.8) 3* You know that  is an acute angle solution to sin x°  a. How can you find other values? [Another value is 180  , then add or subtract multiples of 360 to  and to 180  ]

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31.3 Area of a triangle

31.3 Area of a triangle H Demand Grade

In this section … L DC

M B

H AA*

By the end of this section students will:  Know and be able to use the convention for labelling the sides and angles of a triangle.  Understand and use 12 ab sin C to calculate the area of a triangle.

Prior knowledge Students should already: Know and be able to use area of a triangle  12 base  height. Know how to calculate the length of a side in a right-angled triangle. Be able to find the sine of any angle using a calculator. Checking prior knowledge Non-ICT Starter 31.3 Ask students to explain how they would calculate the area of a triangle. (Use 12 base  height) Draw a right-angled triangle on the board. Ask students to explain how they would calculate the length of any of the sides. Ask students to explain how they would find the sine of any angle using a calculator. TEACHING THE TOPIC

Key vocabulary and phrases Included angle Non-ICT Main Activity 31.3

ICT Main Activity 31.3

(a) Draw a non-right-angled triangle ABC on the board. Ask students to explain how to label the sides and angles according to convention. (See the diagram in the Student’s Book, page 507.) (b) Divide this triangle into two right-angled triangles and label the dividing line h for the height of the triangle. (c) As a class, use trigonometry to express h in terms of the sides of the original triangle and the sine of the appropriate angle. (e.g. h  a sin C.) (d) As a class, find the area of the triangle by using the formula ‘Area  12 base  height’ and substitute the value for h. Show that the area of triangle ABC  12 ab sin C.

Display ICT Main Activity 31.3 which is an animation of finding the area of a triangle  12 ab sin C.

As a class, find the area of a triangle that has sides 7.3 cm and 5.8 cm long with an included angle of 37°. (12.7 cm2) Show by drawing the other two altitudes you can also get the formulae Area  12 bc sin A on Area  12 ac sin B. As a class, work out the included acute angle, if the area of a triangle is 20 cm2 and it has sides of 8.1 cm and 6.4 cm (50.5°). TEACHER’S TIPS

Click ‘Play’ to start, and ‘Pause’ to stop. ‘Restart’ returns the animation to the start. The animation can be manually controlled by dragging the ‘Playback control’. This animation demonstrates where the formula ‘Area of a triangle  12 ab sin C’ comes from. How has the triangle been labelled? (Vertices are labelled with capital letters and the opposite sides with the corresponding small letter.) How would the formula be written using angle A? (Area  12 bc sin A) How would the formula be written using angle B? (Area  12 ac sin B) Which angle do I need when I know the lengths of sides a and c? (Angle B) How could the animation be adapted to give the formula area of triangle  12 ac sin B? (Perpendicular height drawn from vertex A)

 The formula area of triangle  12 ab sin C is given on the examination formula sheet. However, students may be able to remember that the area of a triangle is half the product of the lengths of two sides times the sine of the angle between them.  The derivation of the result 12 ab sin C is not needed for the examination.  When the size of an angle has to be found from the area of a triangle, it is usually expected that only the acute angle is required.

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Individual activity: Exercise 31D Q1 gives practice in finding the area of a triangle when the lengths of two sides and the size of the included angle are given. Q2 requires the area of a quadrilateral to be found by working out the areas of two triangles. Q3 requires the size of the included acute angle to be found when the area is given. Q4 requires the length of a side to be found when the area is given. Q5 involves the recognition that sin °  sin(180  )° Q6 involves finding the area of a regular octagon. Q7 requires students to know that a parallelogram can be formed from two congruent triangles. Q8 involves using the area of an equilateral triangle to work out the area of a regular hexagon. Q9 requires students to remember how to work out the area of a sector and hence to work out the area of a segment. TOPIC SUMMARY

Key questions 1 Here is a parallelogram. What is the area of the parallelogram? [bc sin A]

b A c

2* Here is a regular decagon. An isosceles triangle is formed by joining the ends of a side to the centre of the decagon. 8 cm The equal sides of the triangle are of length 8 cm. (a) Explain why the area of the decagon is given by 320 sin 36° 360° (The angle subtended at the centre of the decagon by the side is   36° 10 The area of the triangle is 12  8  8  sin 36°  32 sin 36° So the area of the decagon is 10  32 sin 36°  320 sin 36°.) (b) Repeat the above but with a regular polygon of n sides and replace the length of 8 cm by a length of 360° r cm. Find a formula in terms of n and r for the area of the polygon. (Area  12 nr2 sin  ) n (c) Explain how this can be used to find an estimate for the area of a circle of radius 10 cm. (Consider a polygon with a large number of sides such as 1000 so put n  1000 and r =10 in this formula. The result is 314.1571983. The area of a circle of radius 10 cm is 314.1592654.)





31.4 The sine rule H Demand Grade

In this section … L DC

M B

H AA*

By the end of this section students will:  Understand, and be able to use, the sine rule in any triangle to find the length of an unknown side and the size of an unknown angle.

Prior knowledge Students should already: Know the convention for naming the angles and sides of a triangle. Know how to use 12 ab sin C, 12 bc sin A, or 12 ac sin B to find the area of triangle ABC. Know and be able to identify an included angle. Know how to use a calculator to find the sine of an angle and to find an angle with a given sine. Checking prior knowledge Non-ICT Starter 31.4 Ask students to explain how to use a calculator to (a) find the sine of an angle (b) find the angle for a given sine. Draw a triangle on the board and label the vertices A, B, C. How should the sides be labelled? (Side opposite A is a, side opposite B is b, side opposite C is c.) You are given the length of sides a and b, which angle do you need to know to be able to find the area? (C) What formula would you use to find the area? (Area  12 ab sin C)

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31.4 The sine rule

TEACHING THE TOPIC

Key vocabulary and phrases Sine rule Non-included angle Non-ICT Main Activity 31.4

ICT Main Activity 31.4

As a class, write down two different ways of finding the area of a triangle ABC using the length of two sides and the sine of their included angle. Then equate the two expressions and, by cancelling, show part of the sine rule. Show how equating the area using a different pair of angles give the remainder of the sine rule. Write the sine rule out in full. a b c      sin A sin B sin C Ask students to explain which measurements they would need to know in order to find a length in a triangle using the sine rule. (e.g. to find a they need b or c and angle A and angle C or angle B.)

Display ICT Main Activity 31.4 which is a simulation showing a triangle ABC with movable vertices. The angles and side lengths (1 d.p.) can be displayed. Side a is opposite angle A, side b opposite angle B, and side c opposite angle C. The buttons under ‘Sine Rule’ can be used to select one of 3 versions of the sine rule.

Ask students what information they would need to know in order to find an angle in a triangle using the sine rule. (e.g. to find angle A they need a and b or c and angle B or angle C.) Find angles A and B to 1 d.p. B 9 cm 52°

A

C

8 cm

Which sides do we know? (b and c) Which angle do we know? (Angle C). Which parts of the sine rule do we b c sin B sin C need?    or    b c sin B sin C Find angle B by substitution. (44.5°) Find angle C by using the angle sum of a triangle. (83.5°)



TEACHER’S TIPS



Drag the vertices to create any triangle. Show angle B, side b and side c. Which version of the sine rule should we use to find b c angle C? e.g.    sin B sin C





Select ‘Sine Rule b and c’ to display. What do we get when we substitute the values for angle B, side b and side c into the sine rule? Drag the values of angle B, side b and side c from the triangle into the equation. This fixes the shape of the triangle. Ask the class to rearrange the equation to make sin C the subject. Click ‘Solve’ to rearrange and begin solving the equation. Clicking on the highlighted part of the equation will evaluate it. What does sin C equal? What does C equal? Click ‘Reset’ and repeat for other triangles to find missing side lengths and angles.

 Students do not need to know the derivation of the sine rule for their GCSE examination. a b c  The sine rule in the form  =  =  is given on the examination formula sheet. sin A sin B sin C sin A sin B sin C The form      which can be used when finding the size of an angle, is a b c not given on the examination formula sheet.  There are six basic ‘parts’ to a triangle, three sides and three angles. For a triangle to be uniquely defined, three of these ‘parts’ must be known, one of which must be a side since knowing three angles does not uniquely define a triangle.  To use the sine rule, one complete fraction and one part of another fraction must be known.  Generally no marks will be awarded in an examination for a scale drawing solution.  To use the sine rule to calculate the length of a side in a triangle, it is necessary to know any two angles and a side, [AAS]. ✓ ✓

? ✓

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 To use the sine rule to calculate the size of an angle in a triangle, it is necessary to know two sides and a non-included angle, [SSA] ? ✓ ✓ ✓

 Students should know that the largest angle in a triangle is opposite the longest side and that the smallest angle is opposite the shortest side. This can be used to check that the size of an answer is sensible.  Using the sine rule to calculate the size of an angle can lead to two solutions and hence two triangles, the ambiguous case. At GCSE it is not necessary to consider this. Individual activity: Exercise 31E This exercise provides basic practice at using the sine rule. Q1 involves finding the length of a side. Q2 involves finding the size of an acute angle. Q3 involves using the sine rule twice in two connected triangles forming a quadrilateral. Q4 and 5 require students to draw a diagram before attempting a solution. Q6 involves bearings. TOPIC SUMMARY

Key questions 1 Which of the following are correct for triangle PQR? r sin P r sin P (a) p   (b) p   sin Q sin R p sin P (c) p sin Q  q sin P (d)     [(b) (c) and (d) are correct] r sin R 2 (a) In order to find the length of the side marked a in this triangle using the sine rule, what must be done first? (The third angle, 71°, must be worked out.) (b) Work out the value of a. (6.58 cm) (c) Why is the size of this answer sensible? (51 71 so a 8 cm and 6.58 is less than 8)

R p

q P

Q

r a

58°

51° 8 cm

sin x sin 37° 3* (a) Sketch a triangle for which     could apply. 37° 10 14 14 (b) Find an acute angle which is a solution to this equation. (57.4°) (c) Find another angle less than 180° which also is a solution to x 14 sin 37° sin x   (122.6°) 10 10 (d) Is there any reason why x could not be 122.6°? (No, 14 10 and 57.4 and 122.6 are greater than 37) (e) What does your answer to (d) mean? (x could be 122.6° as 122.6°  37° 180°, the angle sum of a triangle so that two triangles are possible with the measurements as shown.)

31.5 The cosine rule H Demand Grade

276

In this section … L DC

M B

H AA*

By the end of this section students will:  Understand and be able to use the cosine rule in any triangle to find the length of an unknown side and the size of an unknown angle.

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31.5 The cosine rule

Prior knowledge Students should already: Know the convention for naming the angles and sides of a triangle. Be able to substitute values into a formula involving terms with squares. Know how to use a calculator to find the cosine of an angle and to find an angle with a given cosine. Checking prior knowledge Non-ICT Starter 31.5

ICT Starter 31.5

Ask students to explain how to use a calculator to find (a) the cosine of an angle (b) an angle with a given cosine.

Display ICT Starter 31.5 which shows a series of triangles. Students have to determine the form of the sine rule that will lead to the calculation of the specified side or angle. Students have 10 seconds to give the appropriate form of the sine rule with appropriate values substituted before the answer appears.

Write this formula on the board. h2  a2  b2 Ask students to explain how to find h if a and b are 2   b2) known. ( a Is the negative value relevant in a question about triangles, lengths, etc? (No, a length is always positive.)

Click ‘Play’ to start, and ‘Pause’ to stop. Use the arrows to adjust the time. Use ‘Back’ and ‘Next’ to move back and forth through the questions. ‘Restart’ returns the program to the start. Q1 to 5 ask for a side. Q6 to 10 ask for an angle. Q11 to 15 are mixed. To extend, with a calculator, students could be asked to find the missing dimension.

TEACHING THE TOPIC

Key vocabulary and phrases Cosine rule Non-ICT Main Activity 31.5

ICT Main Activity 31.5

(a) Draw a triangle ABC, similar to the one at the top of p.513 in the Student Book, on the board. Label the vertices A, B and C only and ask students to explain how to label the sides according to convention. (b) Divide the triangle into two from the apex, to form the second diagram on p.507. Add the labels as shown. Also label CN as x and NA as (b  x) (c) As a class, use Pythagoras’ theorem to express h in two different ways in terms of the remaining sides of the original triangle. (c2  x2  h2 a2  (b  x)2  h2) (d) As a class, rearrange the formulae to make h2 the subject and, knowing that h is the same in each equation, equate the two expressions. (c2  x2  a2  (b  x)2 Expand the brackets and simplify. (c2  a2  b2  2bx) How are x and cos C related? (x  a cos C) Substitute for x.

Display ICT Main Activity 31.5 which is a simulation showing a triangle ABC with movable vertices. The angles and side lengths (1 d.p.) can be displayed. Side a is opposite angle A, side b opposite angle B, and side c opposite angle C. The buttons under ‘Cosine Rule’ can be used to select one of 3 versions of the cosine rule. Drag the vertices to create any triangle. Show angle B, side a and side c. If we know the size of angle B, side a and side c what can we use the cosine rule to find? (Side b) Which version of the cosine rule are we going to use? Select ‘Cosine Rule Angle B’ to display. What do we get when we substitute the values for angle B, side a and side c into the cosine rule?

Show that you have derived the cosine rule a2  b2  c2  2bc cos A

Drag the values of angle B, side a and side c from the triangle into the equation. This fixes the shape of the triangle.

In groups, ask the students to find these form of the cosine rule. b2  c2  a2  2ca cos B and c2  a2  b2  2ab cos C

Click ‘Solve’ to begin solving the equation. The equation will rearrange when finding missing angles. Clicking on the highlighted part of the equation will evaluate it.

Ask students which pieces of information they would need to know in order to find a length (e.g. a) in a continued 

What does b equal? continued 

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Non-ICT Main Activity 31.5 (contd)

ICT Main Activity 31.5 (contd)

triangle using the cosine rule. Check that students understand they need to know two sides and the included angle (SAS).

Click ‘Reset’ and repeat for other triangles to find missing side lengths and angles.

As a class, find the length of the side of a triangle opposite angle 117°, which is included between sides of lengths 7.3 cm and 5.8 cm. (11.2 cm)

Show a triangle with 1 side length and 2 angles, 2 side lengths and 1 angle, or 3 side lengths shown. Ask students to find the 3 missing values on the triangle and explain their method. Reveal the missing values.

As a class, rearrange the cosine rule to make the angle the subject of the formula. b2  c2  a2 cos A   2bc Ask students which pieces of information they would need to know in order to find an angle (e.g. A) in a triangle using the cosine rule. Check that students understand they need to know the lengths of all three sides (SSS).





As a class find the angles of a triangle with sides of lengths 12 cm, 10 cm and 15 cm. (41.6º, 85.5º, 52.9º) Use the cosine rule to find the largest angle first, and then the sine rule for the second angle, and ‘angle sum’ for the third angle.) TEACHER’S TIPS

 Students do not need to know the derivation of the cosine rule for the GCSE examination.  The cosine rule in the form a2  b2  c2  2bc cos A is given on the examination formula b2  c2  a2 sheet. The form cos A   which can be used when finding the size of an angle, 2bc is not given on the examination formula sheet. When using this form it is essential to remember that a is the side opposite to A.  The cosine rule only needs to be used once in any triangle.  To use the cosine rule to calculate the length of a side in a triangle, it is necessary to know the other two sides and the angle between these two sides. [SAS]. ?

✓ ✓ ✓

 To use the cosine rule to calculate the size of an angle in a triangle, it is necessary to know all three sides [SSS]. ✓

✓

? ✓

 A common mistake when using the cosine rule is to simplify, for example, 36  16  2  6  4 cos 36°= 52  48 cos 36° as 4 cos 36°.  Care is needed in the calculations when using the cosine rule but the calculations can be done efficiently on a modern scientific calculator. 112  162  132  Calculations such as  also need to be done with care on a calculator. The 2  11  13 use of brackets allows this calculation to be performed efficiently, that is (112  162  132) (2  11  13)  As the cosine of an obtuse angle is negative, there is no ambiguous case when using the cosine rule.  Remind students that if in triangle ABC, a b c, the angle A angle B angle C.

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31.6 Solving problems using the sine rule, the cosine rule and 12 ab sin C

CHAPTER 31

Individual activity: Exercise 31F The questions in this exercise give practice at using the cosine rule. Q1 and 2 involve finding lengths of sides and the sizes of angles. Q3 requires the cosine rule to be used in two connected triangles forming a quadrilateral. Students also have to use 1  ab sin C twice in order to work out the area of the quadrilateral. 2 Q4 requires students to realise that in order to work out the perimeter of a triangle they first have to know the lengths of the sides. Q5 and 6 require students to draw sketches of triangles before they can answer the questions. Q7 involves an isosceles triangle in a circle. The cosine rule does not have to be used in an isosceles triangle but it is as easy as dividing the triangle up into two right-angled triangles. Q8 and 9 involve bearings, with a diagram being given in Q8 but not in Q9. Q10 requires students to work out an angle in a parallelogram. TOPIC SUMMARY

Key questions 1 (a) Write down the cosine rule applied to this triangle in order to work out the size of the angle marked . 52  92  132 63 cos       0.7 259 90 (b) Work out the size of the angle marked . (134.4°)



9 cm

θ



b2  c2  a2 2* In this triangle cos A   2bc Giving a reason in each case, state what is true about angle A when (a) a2 b2  c2 (acute as cos A 0) (b) a2  b2  c2 (equal to 90° as cos A  0) (c) a2 b2  c2 (obtuse as cos A 0)

5 cm

13 cm

A

c b B a C

3* Look at the derivation of the cosine rule. Why is it not mathematically sound to now argue that putting A  90° leads to a2  b2  c2, which proves Pythagoras’ theorem? (Because Pythagoras’ theorem was used in this derivation of the cosine rule.)

31.6 Solving problems using the sine rule, the cosine rule and 12 ab sin C H Demand Grade

In this section … L DC

M B

H AA*

By the end of this section students will:  Be able to identify which of the sine rule and the cosine rule is the appropriate rule to use in solving problems in 2-D and in 3-D.  Be able to use the sine rule and the cosine rule to work out the length of a side or the size of an angle in order to work out the area of a shape.

Prior knowledge Students should already: Know and be able to use the sine rule and the cosine rule. Know and be able to use 12 ab sin C to work out the area of a triangle. Understand and be able to use bearings and the angle between a line and a plane. Know and be able to use properties of special quadrilaterals.

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Checking prior knowledge Non-ICT Starter 31.6 Ask students to explain how to use (a) the sine rule (b) the cosine rule, to find an angle in a triangle. Ask students to tell you the formula for the area of a triangle using the sine of the included angle. Ask students to tell you how to draw a diagram for this journey. I cycle 3 km from home on a bearing of 050°. I stop at a bench for a rest. I cycle another 6 km on a bearing of 120º and stop for lunch. I cycle home, a distance of b km.

N N 3 km

B 120°

50°

A 6 km

b km

N

C

Non-ICT Main Activity 31.6 Look at the journey described in the Starter. How could I find angle B in triangle ABC? (B  50º  60º  110º) What rule could I use to find b? (cosine) As a class, find distance b. (b2  45  36 cos 110°, b  7.57 to 3 s.f.) Draw a triangle on the board and mark one angle and two sides as known. Ask the class which rule (sine or cosine) they should use to find the other side. Repeat with other triangles, using different known angles/sides, and with different angles/sides to find. If the students pick the wrong rule, show them why it is inappropriate.  When using a value that was calculated in one part of a question in a later part of a question, students should be advised and encouraged to use an unrounded value rather than the rounded value given as the answer to the earlier part.  The sine and cosine rules can be used in any triangle, including right-angled triangles. These rules are given as formulae on the examination paper, whereas the trigonometric formulae for a right-angled triangle are not.  Remind students that if in triangle ABC, angle A angle B angle C then a b c

TEACHER’S TIPS

TEACHING THE TOPIC

Key vocabulary and phrases There is no new vocabulary for this section. Use the vocabulary from previous sections. Individual activity: Exercise 31G This exercise requires students to decide which one of the sine rule or the cosine rule to use. Q1 to 3 involve triangles. Q4 uses properties of kites. Q5 involves bearings. Q6 to 8 are 3-D problems. TOPIC SUMMARY

Key questions 1 Look at these triangles. In order to find the length of the marked side or the size of the marked angle, which one of the sine rule or the cosine rule should be used? (You are not expected to work out the length or the size of the angle.) (a) (cosine) (b) (sine) (c) (either) a

8 cm

54°

29° 12 cm

f

7 cm

6 cm

36.18°

14 cm

10 cm 81°

b

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Answers to chapter review questions

2* Try to find the size of the angle marked y°. What happens and why? (No value of y° is possible as cos y°  1.321. This is because 6  7 14 so the triangle cannot exist.)

y° 7 cm

6 cm

14 cm

3* The area of the triangle ABC can be found using the result that abc Area  s (s  a)(  s  b)(  s  c) where s   2 (a) Using this, work out the area of the triangle ABC. (9.80 cm2) (b) Verify this result by working out an angle in the triangle ABC and calculating the area of triangle ABC. (A  101.53°, B  44.42°, C  34.05°, Area  9.80 cm2)

A

5 cm

4 cm

B

7 cm

C

Answers to chapter review questions 1 2 3 4 5

33.60 m a 28.9 cm2 a 56.4 cm2 17.6° a 14.4 cm

6 A(0, 1) B(180, 1) b 9.40 cm b 7.84 cm b 78.5°

7 177 m 8 a 10.1 cm b 13.9 cm2 c 41.3 cm2

9 a i 11.5°, 168.5° ii 216.9°, 323.1° b 78.5°, 281.5° 10 19.5° 11 a 38.2°

b 69.3 cm2

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