Unit Maths Methods
succeeding in the vce, 017 extract from the master class teaching materials Our Master Classes form a component of a highly specialised weekly program, which is designed to ensure that students reach their full potential (including the elite A and A+ scores). These classes incorporate the content and teaching philosophies of many of the top schools in Victoria, ensuring students are prepared to a standard that is seldom achieved by only attending school. These classes are guaranteed to motivate students and greatly improve VCE scores! For additional information regarding the Master Classes, please do not hesitate to contact us on (0) 966 11 or visit our website at www.tsfx.com.au. essential for all year 11 and 1 students! important notes Our policy at TSFX is to provide students with the most detailed and comprehensive set of notes that will maximise student performance and reduce study time. These materials, therefore, include a wide range of questions and applications, all of which cannot be addressed within the available lecture time. Where applicable, fully worked solutions to the questions in this booklet will be handed to students on the last day of each subject lecture. Although great care is taken to ensure that these materials are mistake free, an error may appear from time to time. If you believe that there is an error in these notes or solutions, please let us know asap (admin@tsfx.com.au). Errors, as well as additional advice, clarifications and important updates, will be posted at www.tsfx.com.au/vce-updates. The views and opinions expressed in this booklet and corresponding lecture are those of the authors/lecturers and do not necessarily reflect the official policy or position of TSFX. TSFX - voted number one for excellence and quality in VCE programs. copyright notice These materials are the copyright property of The School For Excellence and have been produced for the exclusive use of students attending this program. Reproduction of the whole or part of this document constitutes an infringement in copyright and can result in legal action. No part of this publication can be reproduced, copied, scanned, stored in a retrieval system, communicated, transmitted or disseminated, in any form or by any means, without the prior written consent of The School For Excellence (TSFX). The use of recording devices is STRICTLY PROHIBITED. Recording devices interfere with the microphones and send loud, high-pitched sounds throughout the theatre. Furthermore, recording without the lecturer s permission is ILLEGAL. Students caught recording will be asked to leave the theatre, and will have all lecture materials confiscated. it is illegal to use any kind of recording device during this lecture
CIRCULAR FUNCTIONS (TRIGONOMETRY) CIRCLES AND TRIGONOMETRIC RATIOS The equation of any circle with centre ( 0, 0) and a radius of r units is x + y = r. OPP sinθ = = HYP y r y y When r = 1: sinθ = = = y i.e. y = sinθ r 1 This means that the y coordinate on the Unit circle is sinθ. ADJ cosθ = = HYP x r x x When r = 1: cosθ = = = x i.e. x = cosθ r 1 This means that the x coordinate on the Unit circle is cosθ. OPP tanθ = = ADJ y x When r = 1: y sinθ tanθ = = x cosθ This means that the y coordinate divided by the x coordinate on the Unit circle is tanθ. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
THE UNIT CIRCLE The equation of a circle with centre ( 0, 0) and radius 1 unit is x + y = r. This equation is known as the UNIT circle, and it is this equation that the principles and definitions in trigonometry are based on. Domain: x 1 Range: y 1 ANGLE MEASURES Angles can be measured in degrees or radians ( c ). The radian is defined as the angle that results when the length of the arc on the UNIT circle is equal to the radius of that circle i.e. 1 One radian Radius of the circle = which is equivalent to approximately 57. o Circumference of the circle As the circumference of a circle is r, there are radians in a full circle. i.e. = 60 o. Therefore, 1 radian o c 180 o ( 1 ) and 1 degree (1 ) = = radians. 180 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
ANGLE CONVERSIONS For example: 4 4 180 0 = = 40 For example: o 160 0 = 160 0 180 c 8 = 9 ANGLE DIRECTIONS The angle measure most commonly used in trigonometry is θ. The value of θ is measured by moving from the positive side of the X axis, in an anticlockwise direction. Positive angles: Negative angles: Move in an anticlockwise direction from the positive X axis. Move in a clockwise direction from the positive X axis. For example: cos = x coordinate at 70 0 = 0 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
The Unit circle is divided into four quadrants: QUADRANTS AND ANGLES The rules describing the angles in each quadrant are: First Quadrant: θ Second Quadrant: θ Third Quadrant: + θ Fourth Quadrant: θ or θ SUMMARY OF SIGNS (CAST) The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 4
ANGLES AT THE AXES nd Quadrant 1st Quadrant rd Quadrant 4th Quadrant If θ is an angle measured anti clockwise from the positive direction of the X axis: sin θ represents the y coordinate of a point P( θ ) on the unit circle. 0 For example: sin = y coordinate at 70 = As the range of the unit circle is y 1 then sin( θ ) 1. cos θ represents the x coordinate of a point P( θ ) on the unit circle. For example: cos = x coordinate at 90 0 = 0 As the domain of the unit circle is x 1 then cos( θ ) 1. tan θ represents the gradient of the radius line that passes through a point P( θ ) on the sinθ unit circle. tanθ = cosθ. For example: 0 y coordinate at 90 tan = 0 x coordinate at 90 = 1 0 = undefined R\ n+ 1, n Z Domain: ( ) As the radius of the unit circle is one, the maximum and minimum values of sin θ and cos θ are ± 1 i.e. All values of sin θ and cos θ must lie between -1 and 1 inclusive. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 5
SUMMARY OF THE EXACT VALUES BASED ON THE AXES Using the Unit Circle, the following exact values may be obtained: sin ( 0) = 0 cos( 0) = 1 tan ( 0) = 0 sin = 1 cos = 0 tan = undefined sin ( ) = 0 cos( ) = 1 ( ) tan = 0 sin = 1 cos = 0 tan = undefined sin ( ) = 0 cos( ) = 1 ( ) tan = 0 EXACT VALUES IN THE FIRST QUADRANT Angles in the first quadrant are referred to as Reference Angles. Therefore, 0 < Reference Angle < Exact values for,, 6 4 are determined by using trigonometric ratios i.e. SOHCAHTOA. sinθ = opposite hypothenuse cosθ = adjacent hypothenuse opposite tan θ = adjacent The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 6
The values of important Reference Angles are given below: Angle ( θ ) 0 6 4 sin θ 0 1 4 cos θ 4 1 0 tan θ 0 1 1 1 Undefined Note: In the first examination paper for Mathematical Methods, the Examiners will assume that the exact values above are known. Learn these values off by heart. SUPPLEMENTARY ANGLES These rules allow you to express trigonometric expressions in terms of: ± acute exact value ± acute exact value By symmetry: o sin(180 θ) = sinθ sin( θ ) = sinθ o sin(180 + θ) = sinθ sin( + θ ) = sinθ o sin(60 θ) = sinθ sin( θ ) = sinθ o cos(180 θ) = cosθ cos( θ ) = cosθ o cos(180 + θ) = cosθ cos( + θ ) = cosθ o cos(60 θ) = cosθ cos( θ ) = cosθ o tan(180 θ) = tanθ tan( θ ) = tanθ o tan(180 + θ) = tanθ tan( + θ ) = tanθ o tan(60 θ) = tanθ tan( θ ) = tanθ cos( θ ) = cosθ sin( θ ) = sinθ tan( θ ) = tanθ The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 7
Exact values for,, 6 4 EXACT VALUES BASED ON TRIGONOMETRIC RATIOS are determined by using trigonometric ratios i.e. SOHCAHTOA. sin θ = cos θ = opposite hypothenuse adjacent hypothenuse opposite tan θ = adjacent Using trigonometric ratios (SOHCAHTOA), the following exact values are obtained: Angle ( θ ) 0 6 4 sin θ 0 1 4 cosθ 4 1 0 tan θ 0 1 1 1 Undefined Note: In the first examination paper for Mathematical Methods, the Examiners will assume that the exact values above are known. Learn these values off by heart. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 8
COMPLEMENTARY ANGLES These rules allow you to write trigonometric expressions in terms of: ± ± acute exact value acute exact value Complementary angles can be found by using trigonometric ratios (SOHCAHTOA). Hypotenuse θ Opposite θ Adjacent To simplify trigonometric expressions with angles ± θ or ± θ (where θ represents an acute angle), we apply the following complementary rules. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 9
EXACT VALUES TO COMMIT TO MEMORY The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 10
GRAPHS OF TRIGONOMETRIC FUNCTIONS THE SINE GRAPH THE COSINE GRAPH The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 11
THE TANGENT GRAPH INVERSE OPERATIONS 1 sin 1 cos 1 tan undoes sin i.e. undoes cos i.e. undoes tan i.e. sin (sin x) cos (cos x) tan (tan x) = x = x = x RECIPROCAL FUNCTIONS 1 secθ = cosθ 1 cosecθ = sinθ 1 cotθ = tanθ The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
SOLVING TRIGONOMETRIC EQUATIONS Step 1: Write all expressions in terms of one trigonometric function. Step : Transpose the given equation so that the trigonometric expression (and the angle) is on one side of the equation, and the constants are located on the other side of the equation. Step : Use the sign in front of the constant on the right hand side to determine the quadrants in which the solutions are to lie (use CAST). Step 4: Calculate the reference angle i.e. the first quadrant solution. If the exact value cannot be determined: Press Inverse Sin, Cos or Tan of the number on the right hand side of the equation (but ignore the sign). For example: Sin ( number on RHS of equation but ignore the sign) (Ensure that the calculator is in Radian Mode). Step 5: Solve for the variable (usually x or θ ). Let the angle equal the rule describing angles in the quadrants in which the solutions are to lie. nd Quadrant Rule: FQA rd Quadrant Rule: + FQA 1st Quadrant Rule: FQA 4th Quadrant Rule: FQA Note: First Quadrant Angle = FQA Let angle = FQA if solution lies in 1 st Quadrant. Let angle = FQA if solution lies in nd Quadrant. Let angle = + FQA if solution lies in rd Quadrant. Let angle = FQA if solution lies in 4 th Quadrant. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
Step 6: Evaluate all possible solutions by observing the given domain. This is done by adding or subtracting the PERIOD to each of the solutions, until the angles fall outside the given domain. For sine and cosine functions: Period = The number in front of the variable For tangent functions: Period = The number in front of the variable Always look closely at the brackets in the given domain and consider whether the upper and lower limits can be included in your solutions. DO NOT discard any solution until the final step. Step 7: Eliminate solutions that do not lie within the specified domain. Note: Students may solve trigonometric equations by rearranging the domain. IMPORTANT POINTS Students may find solutions of trigonometric equations by manipulating the domain. Given an inequation solve the equation without the inequality and then reason from the graph. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 14
QUESTION 58 EXAM 1 Solve the following equation for x : sin x =, x [ 0, 4 ]. Transpose the given equation so that the trigonometric expression (and the angle) is on one side of the equation, and the constants are located on the other side of the equation: sin x = Calculate the reference angle (the first quadrant solution): x = Sin = Use the sign in front of the constant on the right hand side to determine the quadrants in which the solutions are to lie: s are to lie in the quadrants where sine is positive i.e. the 1 st and nd quadrants: S A T C Solve for the variable (usually x ). Let the angle equal the rule describing angles in the quadrants in which the solutions are to lie: x = and x = st Quadrant Angle = = x : x =, Evaluate all possible solutions by adding and/or subtracting the PERIOD to each of the calculated answers observing the given domain: T = 6 = = 1 x : x =,, 7 8, CAS Application: Using a CAS calculator, sketch y = sin x and find the X intercepts. The answers should be the same as the solutions above. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 15
THE SIGNIFICANCE OF SOLVING TRIGONOMETRIC EQUATIONS When solving trigonometric equations, we are locating the points of intersection of the expressions on the left and right hand sides of the equation. For example: Given sin x =, the solution describes: (a) The points of intersection of y = sin x and y =. (b) The values of x on the graph of y = sin x where y =. y 4 1 0 0.5 0 0.5 1.5.5-1 x - - -4 By transposing sin x =, the following equation is obtained: The solution to this equation describes: sin x =. (a) The points of intersection of y = sin x and y =. (b) The values of x on the graph of y = sin x where y =. y 1 0 0.5 0 0.5 1.5.5 x -1 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 16
Note that the solutions to sin x = and sin x = are identical. 4 y 1 0 0.5 0 0.5 1.5.5-1 x - - -4 y 1 0 0.5 0 0.5 1.5.5 x -1 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 17
QUESTION 59 EXAM 1 cos x + 1= 0, x [, 4 ). Solve ( ) Transpose the given equation so that the trigonometric expression (and the angle) is on one side of the equation, and the constants are located on the other side of the equation: Calculate the reference angle (the first quadrant solution): Use the sign in front of the constant on the right hand side to determine the quadrants in which the solutions are to lie: Solve for the variable (usually x ). Let the angle equal the rule describing angles in the quadrants in which the solutions are to lie: Evaluate all possible solutions by adding and/or subtracting the PERIOD to each of the calculated answers observing the given domain: The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 18
QUESTION 60 EXAM 1 Solve sinx+ =, x [, ]. 4 Transpose the given equation so that the trigonometric expression (and the angle) is on one side of the equation, and the constants are located on the other side of the equation: Calculate the reference angle (the first quadrant solution): Use the sign in front of the constant on the right hand side to determine the quadrants in which the solutions are to lie: Solve for the variable (usually x ). Let the angle equal the rule describing angles in the quadrants in which the solutions are to lie: Evaluate all possible solutions by adding and/or subtracting the PERIOD to each of the calculated answers observing the given domain: The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 19
QUESTION 61 EXAM 1 Solve tan + 1 = 0 x, x 0, (, ). The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 0
QUESTION 6 EXAM 1 (VCAA 009) Solve the equation tan( x) = for x,, 4 4 4 4. ( marks) The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
QUESTION 65 EXAM In the following equation, ab, and c are positive constants. The equation a sin( x + b) = c is guaranteed to have at least one solution in the interval 0 x provided only that: A B C c a c a b D b E c 1 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
QUESTION 66 EXAM In the following equation, ab, and c are positive constants. The equation a cos( x b) = is guaranteed to have no solutions in the interval 0 x provided only that: A B C c < a c > a b > D b < E c > 1 c The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
SOLVING COMPLEX TRIGONOMETRIC EQUATIONS Students are required to be able to manipulate expressions in terms of two or more different trigonometric functions, as well as solve questions involving both trigonometric functions and other expressions such as logarithmic, exponential and polynomial functions. To solve expressions written in terms of two or more trigonometric functions, apply one of the following techniques: If the angles are the same: Simplify equations by removing common factors. If the expression is presented in its factorised form (or can be factorised) and is equal to zero, apply the null factor law to obtain solutions. Given both a sine and cosine function write each function on either side of the equality sign. Convert the expression to a tangent function by dividing both sides by cos or sin. Given two or more terms involving the same trigonometric function (but each with different powers) approach the equation as a disguised quadratic (Let A = method). These techniques can only be successfully applied at this level of mathematics if the angles of each of the trigonometric expressions are identical. If the angles are different: Use complementary or supplementary rules to write one angle in terms of the other. Use technology to find the points of intersection. To solve questions involving both trigonometric functions and other expressions such as logarithmic, exponential and polynomial functions: Use technology to find the points of intersection. For example: Solve sin x = e 6x. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 4
QUESTION 70 EXAM 1 Solve each of the following equations for x. (a) sin ( x) sin( x) 0 =, [ 0, ]. (b) cos ( x) cos( x) 1 0 + =, [ 0, ]. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 5
(c) ( cos x + 1)(sin x ) = 0, { x : < x < }. (d) sin ( x) 7sin( x) 4 =, [ 0, ] The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 6
QUESTION 71 EXAM (VCAA 007) x: cos ( x) + cos( x) = 0 = { } A { x:cos( x ) = 0} B C 1 x:cos( x) = 1 x:cos( x) = 1 x:cos( x) = 0 x:cos( x) = 1 1 x:cos( x) = x:cos( x) = D { } E The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 7
QUESTION 7 EXAM Show that the solution to the equation cos( x) = 0.5sin( x), [, ] occurs when x =.6476. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 8
QUESTION 7 EXAM A solution of the equation sin( x ) a cos(x) = 0 is. The value of a is: 6 A B 1 C D E QUESTION 74 EXAM Solve for all values of x given that correct to decimal places. cos(x ) = 5sin x for [ 0, ]. State your answer(s) The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 9
QUESTION 75 EXAM Find the coordinates of the point(s) of intersection of the graphs f( x) = sinx and 6 1 gx ( ) = 0.5e x for [ 0, ]. State your answer(s) correct to decimal places. QUESTION 76 EXAM 15sin(cos( ( t 8))) 0 t Let f() t = a t 10 Find the value of a if f () t is continuous across 0 t 10. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 0
QUESTION 77 EXAM (CHALLENGING QUESTION) Given that cos x = sin 6, < x <, find x without evaluating sin 6. Solve by converting both sides to the same trigonometric function and equating angles. To convert cos to sin write cos x as: sin x or sin + x or sin x or sin + x To convert sin to cos write sin 6 as: cos as cos θ = sinθ or 6 cos + as cos + θ = sinθ or 6 cos 6 cos + 6 as as cos θ = sinθ or cos + θ = sinθ Converting cos to sin gives: sin x = sin 6 As sin( x) = sin( x) then sin = sin 6 6 sin x = sin 6 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
Equating angles: x = 6 x = 6 x = x = 4 6 Period for this expression: x + = 0 6 x = 0 T = = 1 x = 6 This answer is one of a number of possible solutions. To find the remaining solutions, add/subtract the period,. 8 x = + = etc, which falls outside the given domain. Generate a new equation and solve. Continue the process until a solution is obtained across the given domain, < x <. As cos x = sin + x As sin( x) = sin( x) then sin + x = sin 6 Equating angles: + x = 6 x = 6 4 x = = 6 Add/subtract the period,. 4 x = + =, which falls within the given domain. Period for this expression: + x + = 0 6 x + = 0 T = = 1 + x = 6 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
QUESTION 78A EXAM (CHALLENGING QUESTION) Given that cos a= sin b where 0 < b <, find a in terms of b for < a <. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page
QUESTION 78B EXAM 1 Solve sin x+ cosx = 0. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 4
GENERAL SOLUTIONS FOR TRIGONOMETRIC EQUATIONS 1 Period 1 Period 1 Period 1 Period 1 Period 1 0.5 y 0 5 4 0 4 5 x -0.5-1 Once the solutions for the first cycle have been determined, the solutions in subsequent cycles may be calculated by adding and subtracting the period to each answer. If the equation of a circular function has no domain or an infinite domain, there will be an infinite number of solutions. The solutions to such equations are called general solutions and are calculated by applying the rules below: In General: If cos x If tan x If sin x = a then = a then = a then = ± cos ( ) where n Z = + tan ( ) where n Z = + sin ( ) where n Z and a 1. and a R. and a 1. or = ( + 1) sin ( ) where n Z and a 1. Alternative Notation: If sin x = a then n = + ( 1) sin ( ) where n Z and a 1. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 5
Note: To find the inverse trigonometric value, substitute the number and sign on the right hand side of the equation after the equation has been transposed. For example: If cos x = 0.5 then cos ( a) = cos ( 0.5) = Sometimes, the answer given on the CAS will not look the same as what is obtained when equations are solved algebraically. The answers, however, should be equivalent. GENERAL SOLUTIONS FOR COSINE In General: If cos x = a then = ± cos ( ) where n Z and a 1. To find cos ( a) substitute the number and sign on the right hand side of the equation after the equation has been transposed ( a ). For example: If cos x = 0.5 then Therefore, x = n ±, n Z. = cos ( 0.5) =. cos ( a) The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 6
QUESTION 79 EXAM 1 Find the general solution for cos x = 0.5. If cos x a and a 1. cos (0.5) = then = = ± cos ( ) where n Z 6 n± (6n± 1) x= n± = =, n Z Check the answer to this question on the CAS calculator. Are these two sets of answers equivalent? Sometimes the answer given on the CAS will not look the same as what has been calculated, even though both answers, if done correctly, will be equivalent. QUESTION 80 EXAM 1 cos x + 1= 0. Find the general solution to the equation ( ) If cos x = a then = ± cos ( ) where n Z and a 1. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 7
QUESTION 81 EXAM 1 & Show that the general solution for cosx + + = 0, x R. If cos x = a then = ± cos ( ) where n Z and a 1. Note that x in x = n± cos ( a) represents the angle in cos x = a. Therefore, the rule x = n± cos ( a) must be re-written using the actual angle. i.e. x + = n ± cos ( a). cosx + + = 0 cosx + = Base angle: + = ± 1 5 cos = 6 x n cos ( a) 5 x+ = n ± 6 1n ± 5 x + = 6 6 1n ± 5 x = 6 6 1n ± 5 x = 1 1 1n + 5 1n 5 = or 1 1 1n + 1n 7 = or 1 1 (4n+ 1) (1n 7) x= or 1 1 (4n+ 1) (1n 7) x= or where n Z. 4 1 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 8
GENERAL SOLUTIONS FOR TANGENT In General: If tan x = a then = + tan ( ) where n Z and a R. To find tan ( a) substitute the number and sign on the right hand side of the equation after the equation has been transposed ( a ). For example: If tan x = then tan ( a) = tan ( ) = 1.1071 QUESTION 8 EXAM 1 Find the general solution for tan x = 1. If tan x tan (1) = a then = 4 = + tan ( ) where n Z 4n+ (4n+ 1) x= n+ = =, n Z 4 4 4 and a R. Check the answer to this question on the CAS calculator. Are these two sets of answers equivalent? The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 9
QUESTION 8 EXAM 1 Find the general solution for tan x = 1. If tan x = a then Note that x in = + tan ( ) where n Z and a R. = + tan ( ) represents the angle in tan x a Therefore, the rule 1 i.e. x = n+ tan ( a). tan (1) = 4 x= n+ 4 4n+ (4n+ 1) = =, n Z 4 4 (4n + 1) x =, n Z 1 =. = + tan ( ) must be re-written using the actual angle The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 40
QUESTION 84 EXAM 1 Solve tanx + 1 = 0. If tan x = a then = + tan ( ) where n Z and a R. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 41
QUESTION 85 EXAM 1 Solve sinx cosx= 0 for x R. If tan x = a then = + tan ( ) where n Z and a R. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 4
GENERAL SOLUTIONS FOR SINE If the solutions for a are positive: If a is negative: Use Quadrant 4 rules to calculate the basic angle. This means that angles will be negative. If sin x = a then = + sin ( ) where n Z and a 1. or = ( + 1) sin ( ) where n Z and a 1. Alternative Notation: If sin x = a then sin ( a) n = + ( 1) sin ( ) where n Z and a 1. To find substitute the number and sign on the right hand side of the equation after the equation has been transposed ( a ). For example: If sin x = 0.5 then = =. 6 sin ( a) sin ( 0.5) The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 4
QUESTION 86 EXAM 1 Find the general solution for sin x = 0.5. If sin x = a then = + sin ( ) where n Z and a 1. or = ( + 1) sin ( ) where n Z and a 1. = + sin ( ) or x = n+ a ( 1) sin ( ) sin (0.5) = 6 x= n+ or 6 1n+ x = or 6 (1n + 1) x = or 6 x= (n+ 1) 6 6(n + 1) x = 6 (1n+ 6 ) (1n+ 5 ) (1n+ 5) x = = = 6 6 6 Alternatively: If sin x = a then n = + ( 1) sin ( ) where n Z and a 1. Therefore: x = n+ a n x= n + ( 1) 6 n ( 1) sin ( ) Check the answer to this question on the CAS calculator. Are these two sets of answers equivalent? The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 44
QUESTION 87 EXAM (VCAA 009) The general solution to the equation sin( x ) = 1 is: x = n n Z 4 A, B x = n + or x= n, n Z 4 4 n n C x = + ( ), n Z n n D x = + ( ), n Z 4 E x = n + or x= n +, n Z 4 4 If sin x = a then = + sin ( ) or = ( + 1) sin ( ) where n Z. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 45
QUESTION 88 EXAM 1 Find the general solution to the equation sin( x ) = 1. If sin x = a then = + sin ( ) or = ( + 1) sin ( ) where n Z. Alternatively: If sin x = a then n = + ( 1) sin ( ) where n Z and a 1. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 46
QUESTION 89 EXAM 1 & Show that the general solution to sinx+ = is 4 (4n ) x = for n Z. 4 If sin x = a then = + sin ( ) or = ( + 1) sin ( ) where n Z. Alternatively: If sin x = a then n = + ( 1) sin ( ) where n Z and a 1. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 47
QUESTION 90 EXAM 1 Solve sin xcos x= 4cos x, x R. The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 48
QUESTION 91 EXAM 1 5 Solve sinθ = sin, θ R. 7 QUESTION 9 EXAM 1 Solve cos x = cos, x R. 9 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 49
QUESTION 9 EXAM (CHALLENGING QUESTION) Show that the real solutions to the equation sin(x ) = cos x +, x R are given by 4 n x = and x = + n where n Z. 16 8 1 y 0.5 0 0.5 0 0.5 0.5 0.75 1.5 1.5 1.75.5 x -0.5-1 sin(x ) = cos x + 4 Apply the rule sin( θ ) = cos θ to write sin x in terms of cos x : sin( x) = cos x cos x = cos x + Equating angles: x= x+ 4 4 4x = = = 4 4 4 4 x = 16 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 50
General : x = x+ + n where n Z 4 4x = n = n 4 4 4 = n 4 n x = where n Z 16 As cos( θ) = cos( θ) then cos x = cosx+ 4 General : x= x+ + n where n Z 4 + x = x+ + n 4 x = + + n = + + n 4 4 4 = + n 4 x = + n where n Z 8 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 51
SUCCEEDING IN THE VCE 017 UNIT MATHEMATICAL METHODS STUDENT SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/VCE-UPDATES QUESTION 59 QUESTION 60 x = x = 5 11 1,, 4 4 4 5 7,,,,,,, 4 4 4 4 QUESTION 61 QUESTION 6 QUESTION 65 QUESTION 66 1 19 x =,, 1 1 1 x =, 6 Answer is A Answer is B QUESTION 70 (a) (c) x = 0,,, (b) x = 0, x =,, (d) 7 11 x =, 1 1 QUESTION 71 Answer is A QUESTION 7 Solve tan( x) = QUESTION 7 Answer is E QUESTION 74 x = 0.77 QUESTION 75 ( 0.85, 1.854) QUESTION 76 QUESTION 78A QUESTION 78B a = 15sin(1) a = b 5 x = n, + n, + n The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page 1
QUESTION 80 QUESTION 84 QUESTION 85 QUESTION 87 QUESTION 88 QUESTION 89 x = n ± where n Z 4 6 n + (6n+ 1) x = = where n Z 1 1 (6n + 1) x = where n Z 6 Answer is A (4n 1) (4n+ ) x =, where n Z or x= n ( 1) 4 4 or x = n + ( 1) n (4n 1) (n+ 1) 1 n x =, where n Z or x= n ( ) 4 4 4 n QUESTION 90 x = + n or + n where n Z θ = k + 1 where k Z 7 x= k ± where k Z 18 k QUESTION 91 ( ) QUESTION 9 The School For Excellence 017 Succeeding in the VCE Unit Mathematical Methods Page