Download ONTENTS and more Study notes Trigonometry in PDF only on Docsity!
iv
Expressions and Formulae Chapters 1 - 8
Approximation and Estimation...................... 1
Working with Surds.............................. 6
Using Indices.................................. 11
Algebraic Expressions............................ 22
Algebraic Fractions.............................. 33
Gradient of a Straight Line Graph................... 40
Working with Arcs and Sectors..................... 47
Volume of Solids................................ 52
SECTION R EVIEW - E XPRESSIONS AND FORMULAE
Non-calculator Paper............................. 57
Calculator Paper................................ 59
Relationships Chapters 9 - 18
Straight Line Graphs............................. 61
Equations and Inequalities........................ 69
Simultaneous Equations.......................... 81
Formulae...................................... 91
Graphs of Quadratic Functions..................... 98
Quadratic Equations............................ 106
Pythagoras’ Theorem........................... 114
Properties of Shapes............................ 122
Similar Figures................................ 135
Trigonometric Functions......................... 144
SECTION R EVIEW - RELATIONSHIPS
Non-calculator Paper............................ 155
Calculator Paper............................... 157
O N T E N T S C
1
10
SR
SR
SR
SR
7
8
6
5
4
3
2
18
17
16
15
14
13
12
11
9
v
Applications Chapters 19 - 24
Using Trigonometry............................ 159
Vectors...................................... 171
Percentages................................... 182
Working with Fractions.......................... 189
Comparing Distributions......................... 197
Scatter Graphs................................. 207
SECTION R EVIEW - APPLICATIONS
Non-calculator Paper............................ 213
Calculator Paper............................... 215
Exam Practice
Non-calculator Paper............................ 217
Calculator Paper............................... 219
I NDEX....................................... 221
SR
SR
EP
EP
19
20
21
23
22
24
Consider the calculation 600.02 7500.97 4500732. To 1 d.p. it is 4500732.0, to 2 d.p. it is 4500732.02. The answers to either 1 or 2 d.p. are very close to the actual answer and are almost as long. There is little advantage in using either of these two roundings. The point of a rounding is that it is a more convenient number to use.
Another kind of rounding uses significant figures. The most significant figure in a number is the figure which has the greatest place value.
Consider the number 237. The figure 2 has the greatest place value. It is worth 200. So, 2 is the most significant figure.
In the number 0.00328, the figure 3 has the greatest place value. So, 3 is the most significant figure.
(^201)
Expressions and Formulae
Approximation and Estimation
Practice Exercise 1.
1. Write the number 3.9617 correct to (a) 3 decimal places, (b) 2 decimal places, (c) 1 decimal place. 2. The display on a calculator shows the result of 34 7. What is the result correct to two decimal places? 3. The scales show Gary’s weight. Write Gary’s weight correct to one decimal place. 4. Copy and complete this table. 5. Carry out these calculations giving the answers correct to (a) 1 d.p. (b) 2 d.p. (c) 3 d.p. (i) 6.12 7.54 (ii) 89.1 0.67 (iii) 90.53 6. (iv) 98.6 5.78 (v) 67.2 101. 6. In each of these short problems decide upon the most suitable accuracy for the answer. Then calculate the answer. Give a reason for your degree of accuracy. (a) One gallon is 4.54596… litres. How many litres is 9 gallons? (b) What is the cost of 0.454 kg of cheese at £5.21 per kilogram? (c) The total length of 7 equal sticks, lying end to end, is 250 cm. How long is each stick? (d) A packet of 6 bandages costs £7.99. How much does one bandage cost? (e) Petrol costs 133.9 pence a litre. I buy 15.6 litres. How much will I have to pay?
Number 2.367 0.964 0.965 15.2806 0.056 4.991 4. d.p. 1 2 2 3 2 2 2 Answer 2.
68 .kg^847
Rounding using significant figures
To round a number to a given number of significant figures
When rounding a number to one, two or more significant figures:
- Start from the most significant figure and count the required number of figures.
- Look at the next figure to the right of this and l if the figure is 5 or more round up, l if the figure is less than 5 round down.
- Add noughts, as necessary, to locate the decimal point and preserve the place value.
- When answering a problem remember to include any units and state the degree of approximation used.
Noughts which are used to locate the decimal point and preserve the place value of other figures are not significant.
Write 4 500 732.0194 to 2 significant figures. The figure after the first 2 significant figures 45 is 0. This is less than 5, so, round down, leaving 45 unchanged. Add noughts to 45 to locate the decimal point and preserve place value. So, 4 500 732.0194 4 500 000 to 2 sig. fig.
Example 3
Write 0.000364907 to 1 significant figure. The figure after the first significant figure 3 is 6. This is 5 or more, so, round up, 3 becomes 4. So, 0.000364907 0.0004 to 1 sig. fig. Notice that the noughts before the 4 locate the decimal point and preserve place value.
Example 4
What is the area of a rectangle measuring 4.6 cm by 7.2 cm? 4.6 7.2 33. Since the measurements used in the calculation (4.6 cm and 7.2 cm) are given to 2 significant figures the answer should be as well. 33 cm 2 is a more suitable answer.
Example 5
NNoottee: To find the area of a rectangle: multiply length by breadth.
NNoottaattiioonn: Often significant figure is shortened to sig. fig.
Choosing a suitable degree of accuracy
In some calculations it would be wrong to use the complete answer from the calculator. The result of a calculation involving measurement should not be given to a greater degree of accuracy than the measurements used in the calculation.
Practice Exercise 1.
1. Write these numbers correct to one significant figure. (a) 17 (b) 523 (c) 350 (d) 1900 (e) 24. (f) 0.083 (g) 0.086 (h) 0.00948 (i) 0. 2. Copy and complete this table. 3. This display shows the result of 3400 7. What is the result correct to two significant figures? 4. Carry out these calculations giving the answers correct to (a) 1 sig. fig. (b) 2 sig. fig. (c) 3 sig. fig. (i) 672 123 (ii) 6.72 12.3 (iii) 78.2 12. (iv) 7.19 987.5 (v) 124 65300 5. A rectangular field measures 18.6 m by 25.4 m. Calculate the area of the field, giving your answer to a suitable degree of accuracy. 6. In each of these short problems decide upon the most suitable accuracy for the answer. Then work out the answer, remembering to state the units. Give a reason for your degree of accuracy. (a) The area of a rectangle measuring 13.2 cm by 11.9 cm. (b) The area of a football pitch measuring 99 m by 62 m. (c) The total length of 13 tables placed end to end measures 16 m. How long is each table? (d) The area of carpet needed to cover a rectangular floor measuring 3.65 m by 4.35 m.
Number 456 000 454 000 7 981 234 0.000567 0.093748 0. sig. fig. 2 2 3 2 2 3 Answer 460 000
Key Points In real-life it is not always necessary to use exact numbers. A number can be rounded to an approximate number. Numbers are rounded according to how accurately we wish to give details. For example, the distance to the Sun can be given as 93 million miles. You should be able to approximate using decimal places.
You should be able to approximate using significant figures.
You should be able to choose a suitable degree of accuracy.
You should be able to use approximations to estimate that the actual answer to a calculation is of the right order of magnitude.
Write the number using one more decimal place than asked for. Look at the last decimal place and l if the figure is 5 or more round up, l if the figure is less than 5 round down.
Start from the most significant figure and count the required number of figures. Look at the next figure to the right of this and l if the figure is 5 or more round up, l if the figure is less than 5 round down. Add noughts, as necessary, to preserve the place value.
Estimation is done by approximating every number in the calculation to one significant figure. The calculation is then done using the approximated values.
The result of a calculation involving measurement should not be given to a greater degree of accuracy than the measurements used in the calculation.
Review Exercise 1
1. Write these numbers correct to 2 decimal places. (a) 28.714 (b) 6.91288 (c) 12.397 (d) 0.0418 (e) 0. 2. Write these numbers correct to 3 significant figures. (a) 2313 (b) 23.58 (c) 36.97 (d) 503.89 (e) 0. 7. (a) Calculate ^499 .6.7 9 ^130 .0.6 4 (b) Do not use your calculator in this part of the question. By using approximations show that your answer to (a) is about right. 8. Find estimates to these calculations by using approximations to 1 significant figure. Then carry out the calculations with the original figures. Compare your estimate to the actual answer. (a) 7.9 4 .83.9 (b) ^4006 .20.29 (c) ^811 .9.7 140 ..^93 (d) 4.120.^ 096 49. 9. (a) Calculate 0.03^778 ^ .95. (b) Show how you can use approximations to check your answer is about right. 10. Niamh calculates 5 967 000 0.029. She gets an answer of 2 057 586 207. Use approximations to check whether Niamh’s answer is of the right magnitude.
(^201)
Expressions and Formulae
Approximation and Estimation
3. The display shows the result of 179 7. What is the result correct to: (a) two decimal places, (b) one decimal place, (c) one significant figure? 4. Calculate 7.25 0. (a) to 1 decimal place, (b) to 2 decimal places, (c) to 3 decimal places. 5. Calculate 107.9 72.5 (a) to 1 significant figure, (b) to 2 significant figures. 6. Find estimates to these calculations by using approximations to one significant figure. (a) 86.5 1.9 (b) 2016 49. 7. Aimee uses her calculator to multiply 18.7 by 0.96. Her answer is 19.752. Without finding the exact value of 18.7 0.96, explain why her answer must be wrong. 8. Daniel has a part-time job in a factory. He is paid £36 for each shift he works. Last year he worked 108 shifts. Estimate Daniel’s total pay for the year. You must show all your working. 9. Flour costs 78p per kilogram from the flour mill. Rachel bought 306 kg of flour from the mill. She shared the flour equally between 18 people and calculated that each person should pay £1.14. (a) Without using a calculator, show that Rachel’s calculation must be wrong. (b) Roughly how much should Rachel charge each person? Show your working. 10. Estate Agents sometimes quote the floor area of a flat in square metres. They quote an estimate so that buyers can easily compare one flat with another. Write down the lengths and widths of each room to 1 significant figure. (a) Obtain an estimate of the total floor area for each of the two flats. Meadow View Flat Park View Flat Reception 1 4.1 m 6.9 m Reception 1 3.9 m 5.1 m Reception 2 3.9 m 5 m Reception 2 4 m 3.8 m Bedroom 1 3.2 m 3.7 m Bedroom 1 4.1 m 3.9 m Bedroom 2 2.9 m 2.1 m Bedroom 2 3.1 m 2.9 m (b) Work out the actual floor area of each flat. Compare the estimates. 11. By using approximate values, find estimates for these questions. (a) 5.98 3.04 (b) 17.742 9.81 (c) (0.0275)^2 (d) 0.0049 0.000 97 (e) 39.5 0.14 (f) 237.4 38. 12. (a) A rectangular lawn measures 27 metres by 38 metres. A firm charges £9.95 per square metre to turf the lawn. Estimate the charge for turfing the lawn. (b) Is your estimate too large or too small? Give a reason for your answer. 13. The floor of a lounge is a rectangle which measures 5.23 m by 3.62 m. The floor is to be carpeted. (a) Calculate the area of carpet needed. Give your answer to an appropriate degree of accuracy. (b) Explain why you chose this degree of accuracy. 14. 3 5 .4 is approximately equal to 6.
Use this to estimate the value of 7. 59.^12 ^ 1.^3 85 5 .4
