( and 1 degree (1 ) , there are. radians in a full circle. As the circumference of a circle is. radians. Therefore, 1 radian.

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As the circumference of a circle is r, there are radians in a full circle. Therefore, radian o c 80 ( and degree ( ) ) o 80 radians.

1 Angles are usually measured in radians ( c ). The radian is defined as the angle that results when the length of the arc of a circle is equal to the radius of that circle. As the circumference of a circle is r, there are radians in a full circle. Therefore, radian o c 80 ( and degree ( ) ) o 80 radians. 80 Degrees Radians 80 c To convert degrees to radians, multiply by o 80 For example: o c 8 9. o 80 To convert radians to degrees, multiply by c For example: 0 The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

The principles and definitions in trigonometry are based on the UNIT circle which is a circle with centre and a radius of unit. The equation of this circle is x y.

2 The principles and definitions in trigonometry are based on the UNIT circle which is a circle with centre and a radius of unit. The equation of this circle is x y. Domain: Range: (0, 0) x y If is an angle measured anti clockwise from the positive direction of the X axis: sin represents the y coordinate of a point P( ) on the unit circle. 0 For example: sin y coordinateat 70 cos represents the x coordinate of a point P( ) on the unit circle. For example: cos x coordinateat tan represents the gradient of the radius line that passes through a point P( ) that lies sin on the circle tan cos. For example: 0 y coordinateat 90 tan 0 x coordinateat 90 0 undefined The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

3 sin 0 0 sin cos 0 cos 0 tan 0 0 tan undefined sin 0 cos tan 0 sin cos 0 tan undefined cos sin 0 tan 0 Positive angles Move in an anticlockwise direction from the positive X axis. Negative angles Move in a clockwise direction from the positive X axis. For example: cos x coordinateat sec cos cosec sin cot tan As the radius of the unit circle is one, the maximum and minimum values of sin and cos are i.e. All values of sin and cos must lie between - and inclusive. For example: sin does not exist (There is no point on the unit circle where the value of y is ). Tangents can assume values between and. tan any real number. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

sin opposite hypothenuse cos adjacent hypothenuse opposite tan adjacent The values of important Reference Angles are given below: Angle ( ) 0 sin 0

4 Angles in the first quadrant are referred to as Reference Angles. Therefore, 0 Reference Angle Exact values for,, are determined by using trigonometric ratios i.e. SOHCAHTOA. sin opposite hypothenuse cos adjacent hypothenuse opposite tan adjacent The values of important Reference Angles are given below: Angle ( ) 0 sin 0 cos tan 0 0 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 08 The Essentials Mathematical Methods Reference Materials Page

5 To simplify trigonometric expressions with angles or (where represents an acute angle), we apply the following supplementary rules. st Quadrant nd Quadrant rd Quadrant th Quadrant sin cos tan sin( ) sin sin( ) sin sin( ) sin( ) sin cos( ) cos cos( ) cos cos( ) cos( ) cos tan( ) tan tan( ) tan tan( ) tan( ) tan To simplify trigonometric expressions with angles acute angle), we apply the following complementary rules. or (where represents an st Quadrant nd Quadrant rd Quadrant th Quadrant sin cos sin cos sin cos sin cos cos sin cos sin cos sin cos sin The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 5

Note: Complementary angles may also be used to convert both mixtures of trigonometric functions with different angles to the same trigonometric expression with different angles.

For example: sin(x ) cos x cos x cos x cos x cos x Equating angles gives x x x x nt Note: Using complementary and supplementary rules, we can write cos( ) different ways.

6 Note: Complementary angles may also be used to convert both mixtures of trigonometric functions with different angles to the same trigonometric expression with different angles. These expressions can then be solved by EQUATING angles. For example: sin(x ) cos x cos x cos x cos x cos x Equating angles gives x x x x nt Note: Using complementary and supplementary rules, we can write cos( ) different ways. x in cos( x) cos( x) cos( x) cos( x) cos( x) As As As sin cos then cos( x) sin x sin cos then cos( x) sin x sin cos then cos( ) sin i.e. cos( x) sin x As sin cos then cos( ) sin i.e. cos( x) sin x i.e. We can use complementary and supplementary angles to write any trigonometric expressions in different ways. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

Step : Identify the quadrant in which the angle lies.

Step : Write the appropriate quadrant rule and solve.

sin cos tan sin cos tan Use CAST to determine the sign of the answer.

7 Step : Identify the quadrant in which the angle lies. Step : Write the given expression in terms of a st quadrant angle. Step : Write the appropriate quadrant rule and solve. To write an expression whose angle is based on in terms of a first quadrant angle, apply the following rule: sin cos tan sin cos tan Use CAST to determine the sign of the answer. To write an expression whose angle is based on quadrant angle, apply the following rule: in terms of a first sin cos cos sin Use CAST to determine the sign of the answer. For example: cos cos cos cos cos tan tan tan tan tan The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 7

8 By applying the Pythagorean Identity on the Unit Circle, the following relationship is obtained: x y Substituting sin y and cos x, we obtain the following rule: sin cos This statement is true for all values of, and is known as an identity. This identity can be manipulated by dividing every term by either sin or cos. Dividing by sin gives: cot cosec Dividing by cos gives: tan sec These identities may be used to find the value of one trigonometric expression (such as cos) given the value of a different trigonometric expression (such as sin). Note: sec cos cosec sin cos cot sin tan We can also use triangles to help us solve these types of problems. A knowledge of the following triads will assist in the construction of triangles:,, 5 5,, 8, 5, 7 7,, 5 Whichever technique is used, careful consideration must be given to the quadrant in which the solution lies. Make sure that you assign the correct sign (positive or negative) by considering CAST. For example: If Since cos x sin x cosx find sin x where x. sin 9 x sin x 5 Since sin x is positive in nd quadrant: sin x 5 The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 8

9 tan x sin( x) sin xcos x tan x For example: sin xcos x=sin( x) sin(8 x) For example: sin cos sin sin x x x= x = For example: cos( ) cos sin x x x cos x cos sin cos cos sin x For example: tan x tan x tan x tan 8 tan tan 8 tan 8 sin undoes sin i.e. sin (sin x) x cos undoes cos i.e. cos (cos x) x tan undoes tan i.e. tan (tan x) x The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 9

10 Step : 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 : Calculate 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 equationbut ignorethe sign) (Ensure that the calculator is in Radian Mode). Step 5: Solve for the variable (usually x or ). Let the angle equal the rule describing angles in the quadrants in which the solutions are to lie. Note: First Quadrant Angle = FQA Let angle = FQA if solution lies in 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 th Quadrant. nd Quadrant Rule: FQA rd Quadrant Rule: + FQA st Quadrant Rule: FQA th Quadrant Rule: FQA Step : Evaluate all possible solutions by observing the given domain. This is accomplished 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 numberin frontof the variable For tangent functions: Period The numberin frontof 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 across specified domain. Note: Students may also solve trigonometric equations by rearranging the domain. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 0

11 Remember to add/subtract the period to each one of your solutions making sure that you do not exceed domain. To avoid mistakes write each term to the same denominator. Never ever eliminate an answer until the final step (so as to assure that all solutions are obtained). Look closely at brackets around the domain and assess whether the first and last solution can be included. Given a physical/real life situation, pay close attention to the domain. Consider the real life limitations on your solutions. For example - lengths cannot be negative. If number on right hand side ends up being 0 or, use the unit circle to find number of solutions per period. For all other numbers you will get two solutions per period. You cannot find the inverse sine or cosine of a number greater than or less than. i.e. sin number that lies between and + inclusive. cos number that lies between and + inclusive. tan any real number You cannot solve two trigonometric expressions that have different angles algebraically (use technology) unless you can find one expression in terms of the other using complementary rules. You cannot convert an equation containing a sin and cos to tan unless they share the same angle. What does sin x 0. 5 find??? Answer: The points of intersection of y sin x and y What does sin x find??? The X intercepts on the graph of y sin x Given an inequation solve the equation without the inequality and then reason from the graph. When solving questions to a given number of decimal places - make sure the calculator is in RADIAN mode. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

12 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. eg. cos sin cos cos cos sin 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. cos cos sin 0 eg. cos 0 and cos sin 0 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. eg. sin xcosx 0 sin x cosx sin x cosx cosx cosx tan x Given two or more terms involving the same trigonometric function (but each with different powers) apply quadratic factors ( Let A = method). eg. sin (5 x) sin (5 x) Let A sin (5 x) sin (5 x) sin (5 x) A A These techniques can only be successfully applied at this level of mathematics if the angles of each of the trigonometric expressions are identical. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

13 If the angles are different: Use complementary or supplementary rules to write one angle in terms of the other or to write mixtures of trigonometric functions with different angles to the same trigonometric expression with different angles. These expressions can then be solved by EQUATING angles. eg. sin(x ) cos x cos x cos x x x x x nt Otherwise, use the SOLVE or INTERSECT function on your calculator to find solutions. To solve questions involving both trigonometric functions and other expressions such as logarithmic, exponential and polynomial functions: Use the SOLVE or INTERSECT function on your calculator to find solutions. For example: Solve sin x e x. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

14 QUESTION 0 Show that the exact value of 5 sin. Identify the quadrant in which the angle lies: 5 80 o 5 As 5 lies between 80 and 70, the angle lies in the third quadrant. Write the given expression in terms of a st quadrant angle: 5 sin sin Write the appropriate quadrant rule and solve: sin cos tan sin cos tan sin Use CAST to determine the sign of the answer. sin sin sin QUESTION 0 If cos and sin (a) cos (b) sin (c) tan (d) cos sin sin cos, find the exact value of the following expressions: 5 cos sin 5 5 sin 5 cos 5 The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

15 QUESTION 0 Write sin cos in the form acos and hence state the value of a. QUESTION 0 (a) If cos( x) cos and x, find the value of x. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 5

16 (b) If cos( x) cos and Note: x is equivalent to x x, find the value of x.. (c) If tan( x) tan and x, find the value of x. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

17 QUESTION 05 Calculate the exact value of the following expressions: (a) 5 8sin cos (b) sin 7 QUESTION 0 cos Show that sintan. cos The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 7

18 QUESTION 07 cos Write in the form sin a bsin and hence state the value of a and b. QUESTION 08 5 If cos and find 7 sin. Step : Use the appropriate identity to find a solution for the unknown trigonometric expression. 5 As sin cos sin 7 5 sin 89 8 sin sin 89 7 Step : Determine the correct sign by observing the quadrant in which the solution is to lie. Since lies in the second quadrant, sin is positive 8 sin. 7 The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 8

19 QUESTION 09 If tan and find sin and cos. QUESTION 0 5 If cos x, find sin x given that x. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 9

20 QUESTION If sin and find cos and sin. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 0

21 QUESTION Solve sin x, x [ 0, ] across the given domain. 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 first quadrant solution: st Quadrant Angle 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 st and nd quadrants: S A T C Solve for the variable (usually x ). Let the actual angle in the given equation equal the quadrant rules in which the solutions are to lie. Let x Let x x, x, x, 0 Evaluate all possible solutions by observing the given domain. Add and subtract the PERIOD to each of the solutions, until the angles fall outside the given domain: T 5 x : x 0,,,, The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

22 QUESTION Which of the following gives the possible solution(s) to the equation I II III 8 IV sin( x) sin? A B C D E I only II only III only I and II IV only The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

23 QUESTION Find exact solutions of cos x 0 for x [, ]. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

24 QUESTION 5 x Four solutions to cos 0, where a A B C D E a x 7a x, a x 9a x 7a x a x, 5a x a x, a x, a x, a x 7a x a x a x a x and and and and and a a x 9a x a x a x 7a x, are The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

25 QUESTION A solution of the equation sin( x) k cos( x) is. The value of k is: A B C D E 0 QUESTION 7 The equation acos( x b) c, where are positive constants, will not have any solutions in the interval [0, ) provided that: A c a B C b c D a c E b a abc,, The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 5

26 QUESTION 8 Let f ( x) asin(x) and g( x) b, where 0 x and a and b are positive integers. Which of the following statements is not true? A B C D E If If If If If b a 0 b a 0 b a a b a b there are no real solutions to the equation f ( x) g( x). there are solutions to the equation f ( x) g( x). there are solutions to the equation f ( x) g( x). there is solution to the equation f ( x) g( x). there are solutions to the equation f ( x) g( x). The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

27 QUESTION 9 Find exact solutions of tan x for x [, ). The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 7

28 QUESTION 0 Find the smallest positive value of k so that the equation sin( x) k cos( x) will have no solution over the domain [0, 0.]. State your answer correct to decimal places. QUESTION Given that cosx 0.5sin x, x [ 0, ], find the solution(s) for x domain. State your answer(s) correct to decimal places. across the given QUESTION Solve sin x 0.5e x for [ 0, ]. State your answer(s) correct to decimal places. The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 8

29 QUESTION Using algebra, show that the solution to cos x cosx 0, [ 0, ] is and. x x 0 QUESTION Without direct substitution, show that x and sin x 7cos x 0 over the domain [0, ]. x are the exact solutions for The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 9

30 QUESTION 5 CHALLENGING QUESTION Given that cos x sin, x, find x without evaluating sin. Solve by converting both sides to the same trigonometric function and equating angles. To convert cos to sin To convert sin to cos cos as cos as write cos x as: sin x or sin x or sin x or sin x write sin as: cos sin or cos sin or cos cos as as cos sin or cos sin Converting cos to sin gives: sin x sin The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page 0

31 As sin( x) sin( x) then sin sin sin x sin Equating angles: x x x x Period for this expression: x 0 x 0 T x 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 second solution is obtained across the given domain, x. As cos x sin x As sin( x) sin( x) then sin x sin Equating angles: x x x Add/subtract the period,. x, which falls within the given domain. Period for this expression: x x 0 x 0 T The School For Excellence 08 The Essentials Mathematical Methods Reference Materials Page

Unit 3 Maths Methods

Unit 3 Maths Methods 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