2nd Year Maths Higher Level Carl Brien Trigonometry “The secret of getting ahead is getting started.” Mark Twain No part of this publication may be copied, reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from The Dublin School of Grinds. (Notes reference: 2-mat-h-trigonometry). EASTER REVISION COURSES Looking to maximise your CAO points? Easter is well known as a time for students to vastly improve on the points that they received in their mock exams. To help students take advantage of this valuable time, The Dublin School of Grinds is running intensive exam-focused Easter Revision Courses. Each course runs for five days (90 minutes per day). The focus of these courses is to maximise students’ CAO points. Special offer: Buy 1st course and get 2nd course free. To avail of this offer, early booking is required as courses were fully booked last year. What do students get at these courses? EASTER REVISION COURSE FEES: PRICE TOTAL SAVINGS 1st Course €295 €295 - 2nd Course FREE €295 €295 3rd Course €100 €395 €490 4th Course €100 €495 €685 5th Course €100 €595 €880 6th Course €100 €695 €1,075 7th Course €100 €795 €1,270 8th Course €100 €895 €1,465 9th Course €100 €995 €1,660 NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY. FREE DAILY BUS SERVICE For full information on our Easter bus service, see 3 pages ahead. 99 90 minutes of intensive tuition per day for five days, with Ireland’s leading teachers. Oral Preparation Courses 99 Comprehensive study notes. Separate to the Easter Revision Courses, The Dublin School of Grinds is also running Oral Preparation Courses. With the Oral marking component of the Leaving Certificate worth up to 40%, it is of paramount importance that students are fully prepared for these examinations. These courses will show students how to lead the Examiner towards topics that the student is prepared in. This will provide students with the confidence they need to perform at their peak. 99 A focus on simple shortcuts to raise students’ grades and exploit the critically important marking scheme. 99 Access to a free supervised study room. 99 Access to food and beverage facilities. NOTE: These courses are built on the fact that there are certain predicable trends that appear and reoccur over and over again in the State Examinations. ORAL PREPARATION COURSE FEES: PRICE To book, call us on 01-442 4442 or book online at www.dublinschoolofgrinds.ie TOTAL SAVINGS 1st Oral Course €140 €140 - 2nd Oral Course €100 €240 €40 Timetable An extensive range of course options are available over a two-week period to cater for students’ timetable needs. Courses are held over the following weeks: »» Monday 21st March – Friday 25th March 2016 »» Monday 28th March – Friday 1st April 2016 All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel). 6th Year Easter Revision Courses 6th Year Oral Preparation Courses SUBJECT LEVEL DATES Accounting H Monday 21st March – Friday 25th March 8:00am - 9:30am French H Sunday 20th March 10:00am - 2:00pm Agricultural Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm German H Saturday 26th March 10:00am - 2:00pm Applied Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am Irish H Saturday 26th March 10:00am - 2:00pm Art History H Monday 28th March – Friday 1 April 8:00am - 9:30am Spanish H Saturday 19th March 1:00pm - 5:00pm Biology Course A* H Monday 21st March – Friday 25th March 8:00am - 9:30am Biology Course A* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm Biology Course A* H Monday 28th March – Friday 1st April Biology Course B* H Monday 21st March – Friday 25th March 10:00am - 11:30am Biology Course B* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm Biology Course B* H Monday 28 March – Friday 1 April Business H Monday 21st March – Friday 25th March 12:00pm - 1:30pm Note: 5th year students are welcome to attend any 6th year course as part of our buy 1 get 1 free offer. Business H Monday 28th March – Friday 1st April 8:00am - 9:30am 3rd Year Easter Revision Courses Chemistry Course A* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm SUBJECT Chemistry Course B* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm Business Studies H Monday 28th March – Friday 1st April Classical Studies H Monday 21st March – Friday 25th March 8:00am - 9:30am English H Monday 21st March – Friday 25th March 8:00am - 9:30am Economics H Monday 21st March – Friday 25th March 8:00am - 9:30am English H Monday 28th March – Friday 1st April 2:00pm - 3:30pm Economics H Monday 28th March – Friday 1st April French H Monday 28th March – Friday 1st April 12:00pm - 1:30pm English Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm Geography H Monday 28th March – Friday 1st April 12:00pm - 1:30pm English Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am German H Monday 21st March – Friday 25th March 8:00am - 9:30am English Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm History H Monday 21st March – Friday 25th March 4:00pm - 5:30pm English Paper 2* H Monday 28th March – Friday 1st April 10:00am - 11:30am Irish H Monday 28th March – Friday 1st April English Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm Maths H Monday 21st March – Friday 25th March 10:00am - 11:30am French H Monday 21st March – Friday 25th March 10:00am - 11:30am Maths H Monday 21st March – Friday 25th March 12:00pm - 1:30pm French H Monday 28th March – Friday 1st April 8:00am - 9:30am Maths H Monday 28th March – Friday 1st April 10:00am - 11:30am Geography H Monday 28th March – Friday 1st April 8:00am - 9:30am Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm Geography H Monday 28th March – Friday 1st April 10:00am - 11:30am Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm German H Monday 21st March – Friday 25th March 10:00am - 11:30am Science H Monday 21st March – Friday 25th March 2:00pm - 3:30pm History (Europe)* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm Spanish H Monday 21st March – Friday 25th March 12:00pm - 1:30pm History (Ireland)* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm Home Economics H Monday 21st March – Friday 25th March 10:00am - 11:30am 2nd Year Easter Revision Courses Irish H Monday 21 March – Friday 25 March 10:00am - 11:30am SUBJECT Irish H Monday 28 March – Friday 1 April Maths Maths Paper 1* H Monday 21st March – Friday 25th March 8:00am - 9:30am Maths Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm Maths Paper 1* H Monday 28th March – Friday 1st April 10:00am - 11:30am Maths Paper 1* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm Maths Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am Maths Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm Maths Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm Maths Paper 2* H Monday 28th March – Friday 1st April 4:00pm - 5:30pm Maths O Monday 21st March – Friday 25th March 8:00am - 9:30am Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm Physics H Monday 28 March – Friday 1 April 10:00am - 11:30am Spanish H Monday 21st March – Friday 25th March 2:00pm - 3:30pm Spanish H Monday 28th March – Friday 1st April th st th th TIME st 10:00am - 11:30am 8:00am - 9:30am 10:00am - 11:30am th st st 12:00pm - 1:30pm 10:00am - 11:30am Due to large course content, these subjects have been * divided into two courses. For a full list of topics covered in these courses, please see 3 pages ahead. SUBJECT LEVEL DATES TIME 5th Year Easter Revision Courses SUBJECT LEVEL DATES TIME Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am English H Monday 28th March – Friday 1st April 4:00pm - 5:30pm LEVEL LEVEL DATES DATES TIME 8:00am - 9:30am 2:00pm - 3:30pm TIME H Monday 21st March – Friday 25th March 2:00pm - 3:30pm BUY 1ST COURSE GET 2ND COURSE FREE! NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY. BOOK EARLY TO AVAIL OF THE SPECIAL OFFER Trigonometry is worth 5% to 7% of the Junior Cert. It appears on Paper 2. Contents 1) Pythagoras ............................................................................................................................................................................3 3) Minutes................................................................................................................................................................................ 17 2) 4) 5) 6) 7) 8) Sin, Cos, Tan ...................................................................................................................................................................... 11 Angles of elevation/depression................................................................................................................................ 20 From one to another...................................................................................................................................................... 26 Evaluating trigonometric functions........................................................................................................................ 29 Past and probable exam questions ......................................................................................................................... 30 Solutions ............................................................................................................................................................................. 48 © The Dublin School of Grinds Page 2 Carl Brien 1) Pythagoras Pythagoras is a famous rule of trigonometry which comes up every single year in the Junior Cert. On page 16 of The Formulae and Tables Booklet you will see the following: Note: Be careful with the letters a, b and c because the examiner often labels the triangles in your Junior Cert with these letters, but they may have nothing to do with these letters in Pythagoras’ Theorem. The way I like to remember this is: (ππππππ π‘ π πππ)! = (ππ‘βππ π πππ)! + (ππ‘βππ π πππ)! Note: The longest side is always the one across from the right angle. Question 1.1 Find the length of π₯. x 3 4 Comments ©The Dublin School of Grinds Page 3 Carl Brien Question 1.2 Find the length of π¦. y 12 13 Note: Sometimes we are asked to write our answer in ‘surd form’. ‘Surd form’ is just another phrase for ‘square root’, so it just means there will be a square root in our answer. Question 1.3 Find π§, in surd form. 7 4 z Question 1.4 πππ is an isosceles triangle with: ππ = |ππ|, ππ = 72 and β¦πππ = 90°. ind |ππ|. a √72 c b Comments ©The Dublin School of Grinds Page 4 Carl Brien Question 1.5 The lengths of the sides of a right angled triangle are as shown in the diagram. 2x + 2 i) Using the theorem of Pythagoras, write an equation n π₯. x + 1 x + 2 ii) Solve this equation to find π₯ correct to 2 decimal places. Comments ©The Dublin School of Grinds Page 5 Carl Brien Note: Sometimes the examiner gives us a real life situation which requires the use of Pythagoras. You can also be asked to list the assumptions you have made. Don’t let this confuse you. hink bout he ext xample: When fire occurs in high raise buildings, the firefighters cannot use the regular stairs or lifts. They can reach some floors using ladders. In order to determine the ladder length, they apply the Pythagorean Theorem as they can estimate the height of the floor affected and the horizontal distance they can use to keep the ladder in position. Question 1.6 Dublin Fire Brigade use their 10 metre ladder to reach the 3rd storey of an apartment block. The windowsill on the 3rd storey is 8 metres above the ground. Find the distance from the foot of the ladder to the base of the building. Comments ©The Dublin School of Grinds Page 6 Carl Brien Question 1.7 A plane takes 4 hours to travel from Dublin to Madrid at a speed of 575 km/h. The distance from Madrid to Rome is 1300 km. Find the distance from Rome to Dublin correct to the nearest kilometre. List any assumptions you have made. Comments ©The Dublin School of Grinds Page 7 Carl Brien Question 1.8 A kite is held by a string 13 metres long. hen lown n he ind t s etres above the ground. How far is the point A from B, correct to the nearest metre? List any assumptions you have made. Comments ©The Dublin School of Grinds Page 8 Carl Brien Question 1.9 Show that Sutton is 14km further from Birmingham than Walsall is from Birmingham. Walsall Sutton 26km 8km Oldbury 6km Birmingham Comments ©The Dublin School of Grinds Page 9 Carl Brien Question 1.10 The height of a wooden goal post is 2.5m and the width is 7m James wants to play a prank by nailing X bars on the goal as shown: What is the minimum total length of wood that James needs to make the prank a success, correct to the nearest metre? Comments ©The Dublin School of Grinds Page 10 Carl Brien 2) Sin, Cos, Tan On page 16 of The Formulae and Tables Booklet you will see the following: However, as with Pythagoras, the letters a, b and c can be very confusing. Therefore you must learn the rules a different way, as follows: πππ = πππππ ππ‘π βπ¦πππ‘πππ’π π πΆππ = ππππππππ‘ βπ¦πππ‘πππ’π π πππ = The way to remember these formulae is: πππππ ππ‘π ππππππππ‘ SOH CAH TOA or Silly Old Harry, Caught A Herring, Trawling Off America or Some Old Horses, Can Always Hear, Their Owners Approaching Comments ©The Dublin School of Grinds Page 11 Carl Brien To use these formulae, you clearly have to know how to label the sides of a triangle: The hypotenuse is the side across from the right angle. The opposite is the side across from the angle we are using. The adjacent is the other side. hyp x opp or x hyp adj adj opp Note: You should always label your triangle sides. When labelling the sides we can use the nicknames βπ¦π, πππ, πππ. on’t se β, π, π because the letter β often gets mixed up with ‘βπππβπ‘’. Note: In these questions you will often end up with something like πππ π₯ = π₯ = cos !! ! ! Casio: πβπππ‘, πππ ! ! , o o ind π₯ you write . This is called cos inverse. The way you get this on your calculator is by pressing: Sharp: 2ππ πΉπ’πππ‘πππ, πππ If you’re using a Sharp, make sure to include the brackets after the πππ !! . If you’re using a Casio, the brackets are automatically included Jο. You should get 30 as your answer. Note: You need to make sure your calculator is in ‘degrees’ mode (there are other more complicated modes you must use at Leaving Cert but at Junior Cert the examiner can only ask you questions in degrees). To check if your calculator is in degrees mode: Casio: There should be a little ‘D’ at the top of the screen. Sharp: There should be a little ‘DEG’ at the top of the screen. If your calculator isn’t in degrees mode you can change it as follows: Casio: πβπππ‘, ππππ, 3 Comments ©The Dublin School of Grinds Page 12 Carl Brien Sharp: πππ‘π’π, 0, 0 Note: if you have any problems with your calculator make sure to ask me. Your calculator is a sound chap, you just have to know how to deal with him. Question 2.1 Find the measure of the angle x. Question 2.2 In the triangle π΄π΅πΆ, π΅πΆ = 1 and π΄π΅ = 3. Find ∠π΅π΄πΆ . Comments ©The Dublin School of Grinds Page 13 Carl Brien Note: s ell s sing in, os Question 2.3 an o ind angles, e an se hem o ind sides: Find x, orrect o hree ecimal laces. Question 2.4 Calculate the measure of the angle π΅, correct to the nearest degree 9 B 40 Comments ©The Dublin School of Grinds Page 14 Carl Brien Question 2.5 Find π₯, correct to one decimal place. Note: e areful ith our lgebra t he nd f his uestion!!! 17 x 30Λ Question 2.6 An architect’s view of Leinster Rugby’s proposed new North Strand at The RDS is given below: The architect has informed the engineer of the following specifications: |AB|=5m |AC|=15m |β¦DAB|=40Λ Find (i) |AD| correct to two decimal places (ii) |DC| correct to two decimal places Comments ©The Dublin School of Grinds Page 15 Carl Brien Comments ©The Dublin School of Grinds Page 16 Carl Brien 3) Minutes You must be able to change angles from decimals to a thing called ‘minutes’. For example: Change 37.36° to the nearest minute. Casio: Sharp: You should get 37°21′36" To change from decimals to minutes, type in the number with the decimal, then press the button with all the commas (2 buttons above number 8), then press equals. To change from decimals to minutes type in the number with the decimal, then press 2nd function, then DMS. The 37 is the degrees, the 21 is the minutes and the 36 is a thing called seconds. If your value for seconds is 30 or greater you round up. If it is less than 30 you round down. Therefore we will round up: 37°22 Question 3.1 Change 50.27° to the nearest minute. Question 3.2 Change 50.38° to the nearest minute. Comments ©The Dublin School of Grinds Page 17 Carl Brien Question 3.3 Change 60.73° to the nearest minute. Question 3.4 Change 12.39° to the nearest minute. Sometimes an angle will be given to you in degrees and minutes so you have to be careful when putting this into your calculator. For example: Find sin 31°26′ correct to 4 decimal places. Casio Calculator: Sharp Calculator: Press Sin, hen 1, hen he omma utton, hen 26, then the comma button again, then equals. Press Sin, then 31, then the DMS button, then 26, then the DMS button again, hen quals. You should get 0.5215061194 (depending on your calculator you may even get more decimal places). We are asked to round off to 4 decimal places so our answer is 0.5215 Question 3.5 Evaluate sin 35°7′ to four decimal places. Question 3.6 Comments ©The Dublin School of Grinds Page 18 Carl Brien Evaluate cos 40°8′ to four decimal places. Question 3.7 Evaluate tan 52°29′ to three decimal places. Question 3.8 Evaluate sin 12°98′ to three decimal places. Question 3.9 Evaluate cos 82°33′ to four decimal places. Comments ©The Dublin School of Grinds Page 19 Carl Brien 4) Angles of elevation/depression There are two other points we must know to help us with ‘real life’ scenarios: i) Angle of elevation: This is the angle your line of vision makes with the horizontal when looking up: Li ig of s ne ht Angle of elevation Horizontal ii) Angle of depression: This is the angle your line of vision makes with the horizontal when looking down: Horizontal Line of s ig Angle of depression ht Comments ©The Dublin School of Grinds Page 20 Carl Brien Question 4.1 A boat is anchored off the coast near 100m high cliffs in Co. Kerry. The angle of elevation from the boat to the cliff top is 15Λ. he ext day the boat moves further towards land and anchors again. This time the angle of elevation is 38Λ. ow far did the boat move? Give your answer correct to the nearest metre and make assumptions as necessary, stating them clearly. Comments ©The Dublin School of Grinds Page 21 Carl Brien Question 4.2 A boat sails due east from the base, A, of a 30m high lighthouse, [AD]. t he oint B the angle of depression of the boat from the top of the lighthouse is 68°. en econds ater he oat s t he oint C and the angle of depression is now 33°. i) ii) Find |BC|, he istance he oat as ravelled n his ime. Calculate the average speed at which the boat is sailing between B and C. ive our nswer n second, correct to one decimal place. D etres er 30m A B C Comments ©The Dublin School of Grinds Page 22 Carl Brien Question 4.3 The beach at Brittas Bay slopes downwards at a constant angle of 12Λ to the horizontal. How far out into the sea can a man walk before the water covers his head? Make assumptions as necessary. Comments ©The Dublin School of Grinds Page 23 Carl Brien Note: If you’re asked to “explain your answer”, simply explain it with maths. For example, in the question below the answer is Yes or No. When they ask you to explain your choice, don’t waffle. Explain using methods like Pythagoras, sin, cos, tan etc. Question 4.4 The city council decides to construct a ramp for the local skate park. Before they start construction, they ask some skaters for their advice: “We want a ramp that is 3 meters high at the top, but we don’t want the angle of elevation to be more than 20Λ.” The city council comes up with a design. The design has the ratio of max height to ramp length of 3:8. Does this designs fit in with the skaters requests? Explain your answer. Comments ©The Dublin School of Grinds Page 24 Carl Brien Question 4.5 A builder called Mark wants to build a canopy roof. Since he is worried about the chance of collapse due to the weight of snow, he asks the advice of a friend in Canada who is in the building trade. His Canadian friend says that it is best to build a roof with a pitch greater than 40Λ so that the snow will not accumulate. Mark’s design for the roof is illustrated below: 3.5 m 8 m Has Mark followed the advice of his friend? Explain your answer. Comments ©The Dublin School of Grinds Page 25 Carl Brien 5) From one to another We can be given what πππ, πΆππ or πππ is equal to and then from this be asked to find another one. There are 5 steps: 1) Sketch a right angled triangle. 2) Label the angles given. 3) Label the two sides given. 4) Find the missing side using Pythagoras. 5) Fill out the answer. Question 5.1 If π πππ₯ = ! ! , find the value of cos π₯. Comments ©The Dublin School of Grinds Page 26 Carl Brien Question 5.2 Given that Tan A = 4, write CosA in the form ! ! , x ∈ N. Question 5.3 ! ! ! ! ! Given that Cos x = , write Sin x in the form , a, b ∈ N Comments ©The Dublin School of Grinds Page 27 Carl Brien Sometimes they actually give you the triangle, which is handy Jο. Question 5.4 Use the information given in the diagram to find Sin A and Cos A. ive our nswer n urd orm. 7 A 2 Comments ©The Dublin School of Grinds Page 28 Carl Brien 6) Evaluating trigonometric functions This section sounds complicated but these types of questions are actually embarrassingly easy. You simply use page 13 of the Formulae and Tables booklet, or your calculator. Note: If given Sin! θ, we rewrite this as (Sin θ)! . The same rule applies for Cos and Tan. f sing our calculator, you must put in the brackets! Question 6.1 Evaluate: Sin! 30Λ + Tan! 45Λ. Question 6.2 Evaluate: 1 − 2πππ! 45°. Question 6.3 Verify that: !"#!"°!"#!"° ! ! !"# ! !"° =4 Comments ©The Dublin School of Grinds Page 29 Carl Brien 7) Past and probable exam questions Question 1 ©The Dublin School of Grinds Page 30 Carl Brien Question 2 ©The Dublin School of Grinds Page 31 Carl Brien Question 3 a) b) c) d) (More space for your solution on next page) ©The Dublin School of Grinds Page 32 Carl Brien ©The Dublin School of Grinds Page 33 Carl Brien Question 4 ©The Dublin School of Grinds Page 34 Carl Brien Question 5 ©The Dublin School of Grinds Page 35 Carl Brien Question 6 ©The Dublin School of Grinds Page 36 Carl Brien Question 7 ©The Dublin School of Grinds Page 37 Carl Brien Question 8 ©The Dublin School of Grinds Page 38 Carl Brien ©The Dublin School of Grinds Page 39 Carl Brien Question 9 ©The Dublin School of Grinds Page 40 Carl Brien ©The Dublin School of Grinds Page 41 Carl Brien Question 10 ©The Dublin School of Grinds Page 42 Carl Brien Question 11 ©The Dublin School of Grinds Page 43 Carl Brien Question 12 ©The Dublin School of Grinds Page 44 Carl Brien Question 13 ©The Dublin School of Grinds Page 45 Carl Brien Question 14 ©The Dublin School of Grinds Page 46 Carl Brien Question 15 (Note: this question contains a mixture of topics) ©The Dublin School of Grinds Page 47 Carl Brien 8) Solutions Question 1.1 x2 = 42 + 32 x2 = 16 + 9 x2 = 25 x = 5 Question 1.2 132 = 122 + y2 169 = 144 + y2 169 144 = y2 25 = y2 βΉy=5 Question 1.3 7 2 = 4 2 + z2 49 = 16 + z2 49 – 16 = z2 33 = z2 βΉ z = 33 Question 1.4 ( 72 )2 = |ac|2 + |bc|2 72 = 2π₯ 2 (Remember: |ac| = |bc|) !" = x2 ! 36 = x2 36 = x = 6 Question 1.5 (2x+2)2 = (x+2)2 + (x+2)2 1) (2x+2)(2x+2) = 4x2 + 4x + 4x + 4 = 4x2 + 8x + 4 2) (x+1)(x+1) = x2 + 2x + 1 3) (x+2)(x+2) = x2 + 4x + 4 ∴ 4x2 + 8x + 4 = x2 + 2x + 1 + x2 + 4x + 4 4x2 + 8x + 4 = 2x2 + 6x + 5 2x2 + 2x 1 a = 2, b = 2, c = 1 !!± ! ! !!!" !! = = = !!± !! !!(!)(!!) !(!) !!± !! !!(!)(!!) !(!) !!± !!! !!± !" ! !!! !" = ! !!± !" x= or x= ! ! x = 0.37 or x= 1.37 ©The Dublin School of Grinds Page 48 Carl Brien Question 1.6 (10)! = π₯ ! + (8)! 100 = π₯ ! + 64 100 − 64 = π₯ ! 36 = π₯ ! 36 = π₯ 6 π = π₯ Question 1.7 (π₯)! = 2300 ! + (1300)! π₯ ! = 5,290,000 + 1,690,000 π₯ ! = 6,980,000 π₯ = 6,980,000 π₯ = 2641.968963 ππ x = 2642 km Question 1.8 Assume that the kite (length of string) is at its longest which is 13 m. 132 = 92 + |AB|2 169 = 81 + |AB|2 88 = |AB|2 |AB| = 88 |AB=9m| Question 1.9 Find distance from Walsall to Birmingham hyp2 = 82 + 62 βΉ hyp2 = 64 + 36 βΉ hyp2 = 100 βΉ hyp = 10 Now find distance from Sutton to Birmingham 262 = 102 + x2 676 = 100 + x2 676 – 100 = x2 x2 = 576 x = 576 = 24 Sutton is 14km further from Birmingham than Walsall is from Birmingham. ©The Dublin School of Grinds Page 49 Carl Brien Question 1.10 hyp2 = 72 + 2.52 hyp2 = 49 + 6.25 hyp2 = 55.25 hyp = 55.25 hyp = 7.433 We need two planks so we need to multiply 7.433 by 2 = 14.866. The minimum length, correct to the nearest metre, is 15m. Question 2.1 ! x = sin 1( ) = 30o ! Question 2.2 ! |∠π΅π΄πΆ| = tan 1 ( ) = 30o ! Question 2.3 ! cos 45o = !" 40 cos 45o = x x = 28.284 ©The Dublin School of Grinds Page 50 Carl Brien Question 2.4 !" B = tan 1( ) = 77o ! Question 2.5 tan 30o = x= Question 2.6 (i) Find |AD| (ii) Find |DC| x= !" !"# !" !" ! = 29.4 Cos 40o = ! !"# !" ! ! = 6.53m First get |DB|: 6.532 = 52 + x2 42.6409 = 25 + x2 17.6409 = x2 x = 4.2m |DC|2 = 102 + 4.22 |DC|2 = 117.6409 |DC| = 117.6409 |DC| = 10.85 ©The Dublin School of Grinds Page 51 Carl Brien Question 3.1 50.27° to the nearest minute = 50° 16’ Question 3.2 50.38° to the nearest minute = 50° 23’ Question 3.3 60.73° to the nearest minute = 60° 44’ Question 3.4 60.73° to the nearest minute = 60° 44’ Question 3.5 Sin 35° 7’ = 0.5752 Question 3.6 Cos 40° 8’ = 0.7645 Question 3.7 Tan 52° 29’ = 1.302 Question 3.8 Sin 12° 98’ = 0.236 Question 3.9 Cos 82° 33’ = 0.1297 Question 4.1 Tan 15° = !"" !"" ! Tan 35° = !"" !"" ! y= x= !"# !"° !"# !"° x = 373.21m y = 128m π·ππ π‘ππππ πππ£ππ = π₯ − π¦ = 373.21 − 128 = 245π Question 4.2 (i) ©The Dublin School of Grinds Page 52 Carl Brien !" Tan 68° = ⇒ |AB| = |!"| !" !"# !"° = 12.121m !" Tan 33° = ⇒ |AC| = |!"| !" !"# !!° = 46.196m ∴ BC = AC – AB = 34.075 m (ii) π΄π£πππππ πππππ = 34.075π = 3.41 π/π 10 π ππ Question 4.3 If the man is Carl, assume his height to be 1.5 m. x is the distance Carl needs to walk out. !.! Tan 12° = ⇒x= ! !.! !"# !"° = 7.1m Question 4.4 Ramp height =3m ! θ = Sin 1( ) = 22° ! No, as 22° > 20° Question 4.5 ©The Dublin School of Grinds Page 53 Carl Brien !.! θ = Tan 1( ) = 41.2° ! He has followed the advice, as 41.2° > 40° ©The Dublin School of Grinds Page 54 Carl Brien Question 5.1 sin x = ! ! 42 = ( 7)2 + adj2 16 = 7 + adj2 9 = adj2 adj = 3 ⇒ Cos x = !"# !!" = ! ! Question 5.2 x2 = 42 + 12 x2 = 17 x = 17 ! ⇒ Cos A = !" Question 5.3 72 = 12 + opp2 49 = 1 + opp2 opp2 = 48 opp = 48 = 4 3 Sin x = ©The Dublin School of Grinds ! ! ! Page 55 Carl Brien Question 5.4 x2 = 72 + 22 x2 = 49 + 4 x2 = 53 x = 53 ! Sin A = Cos A = !" ! !" Question 6.1 Sin2 30° + Tan2 45° = =1 ! ! ! ! +1 Question 6.2 1 – 2 Tan2 45° = 1 – 2 = –1 Question 6.3 Verify that ! ©The Dublin School of Grinds !"# !"° !"# !"°!! !"# ! !"° ! !! ! ! ! = ! ! ! =4 =4 Page 56 Carl Brien Solutions to past and probable exam questions Question 1 (a) |!"| Sin 60° = !" 10(Sin 60°) = |AB| |AB| = 8.66 cm (b) (i) |!"| Tan 65° = !" 10 Tan 65° = |op| |op| = 21.4 cm (ii) |∠ mon| = |∠ mop| + |∠ pon| |∠ mop| = 25° !".! |∠ pon| = Cos 1( ) = 44.49° !" 44.49° + 25° = 69.5° Question 2 (i) (ii) Sin 30° = ! ! ! ⇒x= !"# !"° x=4 Use Pythagoras: 42 = 22 + adj2 16 = 4 + adj2 12 = adj2 adj = 12 = 2 3 = 3.464m 3.464 ×2 = y y = 6.93m ©The Dublin School of Grinds Page 57 Carl Brien Question 3 (a) Tan A = 4 ⇒ opp = 4 and adj = 1 x2 = 42 + 12 x2 = 16 + 1 x2 = 17 x = 17 ! Cos A = (b) !" !"# ! = Cos = !!" ! 2 π₯ = 3 9 x= (c) !(!) ! =6 x2 = 72 + 22 x2 = 49 + 4 x = 53 Sin A = Cos A = (d) ! !" ! !" 32 = 22 + x2 9 = 4 + x2 x2 = 5 x= 5 Sin θ = ©The Dublin School of Grinds ! ! Page 58 Carl Brien Question 4 (i) |ac|2 = 92 + 122 |ac|2 = 81 + 144 |ac| = 225 = 15 (ii) Tan 36.87° = ⇒ |cd| = (iii) !" !"# !".!"° !" |!"| = 20ππ Find |ad| |ad|2 = 152 + 202 |ad|2 = 225 + 400 |ad|2 = 625 |ad| = 25km Total length = 25 + 20 + 12 + 9 = 66km Question 5 (i) Tan 1 (ii) !.! ! Tan 28° = = 28° ! !" 13(Tan 28°) = x x = 6.9m Question 6 (a) Sin 25° = 0.423, Cos 25° = 0.906, Tan 25° = 0.466 Sin 50° = 0.766, Cos 50° = 0.643, Tan 50° = 1.192 (b) (i) No. While Sin and Cos are always less than 1, tan is only less than 1 for angles less than 45°. (At 45°, opp = adj ⇒ Tan 45° = 1, Tan 50° = 1.192) (ii) No. Trigonometric functions are not linear. !"# !"° E.g. = 1.8 ≠ 2 !"# !"° (iii) ∴ Cos 40° < Cos 20° No. Cos reduces with increasing angle (for θ < 180°) Cos 40° = 0.766, Cos 20 = 0.940 ©The Dublin School of Grinds Page 59 Carl Brien Question 7 (a) ! Tan 60° = ⇒ 3 x = h ! (b) Tan 30° = 1 ! ! !"!! 3 x 24 −π₯ 3 24 – x = 3x 24 = 4x x=6 = h = 3(6) = 10.39m Question 8 (a) Angle of elevation = 65° (b) ! Tan 65° = !" h = 10(Tan 65°) = 21.45m (c) Total height of tree = 21.45m + 1.75m = 23.2m (d) He may not be as tall as Sean Question 9 (a) (i) (ii) 24 Tan A = (b) (i) !" ! Cos 60° = (ii) ! ! |pr| : Cos 60° = |!"| ! !" ⇒ 12( ) = |pr| = 6 ! (c) (i) |∠ bac| = 41° alternate angles (ii) Tan 41° = ⇒ |ba| = ©The Dublin School of Grinds !" |!"| !" !"# !"° = 40.26 Page 60 Carl Brien Ans. = 40m Question 10 (a) (b) Sin 1 (0.5045) = 30.2982° |!"| Cos 50°48’ = !" 20(Cos 50°48’) = |rq| |rq| = 12.6m (c) Tan Question 11 (a) (b) 1 ! = = 50° ! |AB|2 = |BC|2 + |AC|2 22 = 12 + |AC|2 4 = 1 + |AC|2 |AC|2 = 3 ⇒ |AC| = 3 Cos ∠ BAC = (c) ! ! = 30° x2 = 12 + 12 x2 = 2 x= 2 ! Cos 45° = ! (d) Cos 75° = !! ! Cos 45° = Cos 30° = ! ! ! ! ! = 0.259 = 0.707 = 0.866 0.259 ≠ 0.707 + 0.866 ©The Dublin School of Grinds Page 61 Carl Brien Question 12 (a) The circumference will be used to calculate the radius of the Spire, which will help to give the full distance that Maria is from the Spire. 2πr = 7.07m r= Tan 60° = !.!" !! = 1.126 ! !".!"# = 123.194 m Add Maria’s height, 123.194 + 1.72 Ans = 125m ©The Dublin School of Grinds Page 62 Carl Brien Question 13 (a) |∠ B| = 50° Sin B = 0.766 (b) ! Sin 50° = ! ⇒ 8(Sin 50°) = h H = 6m (c) ! Area of ABC = Base × Height ! ! = (7.5 × 6) = 22.5 cm2 ! Question 14 (a) (b) ! Tan 30° = !" ⇒ x = 18 (Tan 30°) = 10.4m We need to know what Jasmine’s height is and add it to the result we got in part (a). ©The Dublin School of Grinds Page 63 Carl Brien Question 15 (i) Use construction 13 on page 67 of the notes. (ii) 530 (iii) cos 53° = 0.60181502 = 0.602 (correct to 3 decimal places) (iv) Because the answer in part (ii) is rounded off to a whole number it gives a different answer. ©The Dublin School of Grinds Page 64 Carl Brien
