# FURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C4 Edexcel: C3 OCR: C3 OCR MEI: C4 FURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS Version :. Date: Examples -4, 8-3 and 7-8 are copyrighted to their respective owners and used with their permission.

2 Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 Further Trigonometric identities. The Pythagorean Identities (revision. For all angles cos + sin = (This is the Pythagorean identity tan sin cos From the definitions of the reciprocal functions of sin x = cosec x (the cosecant of x, sometimes abbreviated to csc x cosx tan x = sec x (the secant of x = cot x (the cotangent of x it is possible to obtain several new identities. Because the cotangent is the reciprocal of the tangent, we have cot cos sin By dividing the Pythagorean identity cos + sin = throughout by cos we have + tan sec A similar division of the identity cos + sin = throughout by sin gives cot cosec

3 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 3 of 7 Compound Angle Identities. The following identities hold true for all angles A and B. sin (A + B = sin A cos B + cos A sin B. cos (A + B = cos A cos B - sin A sin B. sin (A - B = sin A cos B - cos A sin B. cos (A - B = cos A cos B + sin A sin B. The formula for tan (A +B can be worked out as sin Acos B cos Asin B sin( A B sin Acos B cos Asin B tan( A B cos Acos B cos Acos B cos( A B cos Acos B sin Asin B cos Acos B sin Asin B cos Acos B cos Acos B This formidable expression simplifies to tan (A + B = tan A + tan B tan A tan B The expression for tan (A -B can be obtained similarly. tan (A - B = tan A - tan B + tan A tan B In each case the results for (A-B can be derived from those for (A+B by reversing the + and - signs a useful revision hint!

4 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 4 of 7 Double and Half Angles. By taking the compound angle formulae and replacing B with A, we obtain the double angle identities. sin (A + A = sin A cos A + cos A sin A = sin A cos A. cos (A + A = cos A cos A - sin A sin A = cos A - sin A. tan (A + A = tan A + tan A = tan A tan A tan A -tan A The formula for cos A can be written in two other useful variant forms by using cos A+ sin A=. cos A - sin A = cos A (- cos A = cos A. cos A - sin A = (-sin A - sin A = - sin A. The double angle formulae are thus: sin (A = sin A cos A cos (A = cos A - sin A = cos A = - sin A tan (A = tan A -tan A By replacing A with ½A in the last two formulae for cos A, we have the half angle identities: cos A = cos ½A cos ½A = + cos A cos ½A = ½( + cos A cos A = - sin ½A -sin ½A = cos A - sin ½A = - cos A sin ½A = ½( - cos A The above identities are useful in further calculus, especially in integration.

5 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 5 of 7 Using and Proving Trigonometric Identities. The above identities can be manipulated to give rise to new ones. Using and proving identities is a skill which improves with practice. Example (: Find the values of sin 75 and cos 75, leaving the results as surds. From the results, also find the values of tan 75 and tan 5. (Note that tan 5 = cot 75 We treat 75 as and apply the identity sin ( = sin 45 cos 30 + cos 45 sin 30. Since sin 45 = cos 45 = or ( 3 or 3., the expression for sin 75 becomes (cos 30 + sin 30 To find cos 75, we substitute into the identity cos ( = cos 45 cos 30 - sin 45 sin 30. This expression simplifies into or ( 3 or 3. (cos 30 - sin 30 To find tan 75, we could either divide the result for cos 75 into the one for sin 75, or we could substitute into the compound angle tangent formula. The question asks for the former method. Dividing sin 75 by cos 75 we have tan 75 = 3 3 = = = =. (rationalising the denominator To find tan 5, we can divide the result for sin 75 into cos 75, or find the reciprocal of tan 75. = = = =. (rationalising the denominator

6 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 6 of 7 cos A cos A sin A Example (: Prove that cos A sin A. (Copyright Letts Educational, Revise AS and A Mathematics (004 ISBN , Example Proving Identities p.33 The first step is to start with the expression on one side of the identity, and then to replace parts of it with an equivalent form. The LHS is more complex, and so we can rewrite it using the double angle result for cos A. The LHS thus becomes two squares, and so it can be rewritten cos A sin A cos A sin A (cos A sin A(cos A sin A (cos A sin A, but then the top line can be recognised as the difference of After taking out cos A sin A as a factor, the expression becomes cos A + sin A, thus LHS = RHS.. Example (3: Prove that csc sec csc A sin A A A (Copyright Letts Educational, Revise AS and A Mathematics (004 ISBN , Example Proving Identities, p.33 (The software on the computer uses csc for cosec. Since cosec A = sin A, we can multiply both top and bottom of the LHS by sin A to give sin, which is equivalent to A cos A (using cos A +sin A =, and finally to sec A as on the RHS (using sec A = cosa. Example (4: Prove that sin tan tan. (Copyright Letts Educational, Revise AS and A Mathematics (004 ISBN , Trigonometric Equations, Progress Check Ex., p.34 Rewrite the RHS as tan sec (using + tan sec tan cos (using sec = cos sin cos cos (using tan = sin cos sin cos, which is the same as sin LHS = RHS

7 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 7 of 7 Example (5: Use the result of the previous example to express cos A in terms of tan A. Since sin A tan A, it follows that cos A sin A cos A tan A cos A sin A tan A. tan A tan A cos A tan A tan A tan cos A tan A A (previous example plus double angle formula These examples are also often written as half-angle identities involving tan ½ If t = tan ½then the other ratios for can be obtained in terms of t, as per the diagram. The standard double-angle formula is t tan t Using Pythagoras, the length of the hypotenuse is t ( t From this result, it follows that 4 4t t t 4 t t t. sin t t and cos t t.

8 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 8 of 7 Example (6: Show that sin P + sin Q = sin ½(P+Q cos ½(P-Q. Hint: use P = A + B and Q = A - B Take the two compound angle sine identities and add them together: sin (A + B + sin (A - B = (sin A cos B + cos A sin B + (sin A cos B - cos A sin B = sin A cos B. Substituting P = A + B and Q = A B, we have A = ½(P+Q and B = ½(P-Q. The summed expression above therefore becomes sin P + sin Q = sin ½(P + Q cos ½(P - Q. Had we subtracted the second sine identity from the first, we would have had sin (A + B - sin (A - B = (sin A cos B + cos A sin B - (sin A cos B - cos A sin B = cos A sin B, or sin P - sin Q = cos ½(P + Q sin ½(P - Q. The cosine identities are treated similarly. cos (A + B + cos (A - B = (cos A cos B - sin A sin B + (cos A cos B + sin A sin B = cos A cos B. cos P + cos Q = cos ½(P + Q cos ½(P - Q. cos (A + B - cos (A - B = (cos A cos B - sin A sin B - (cos A cos B + sin A sin B = - sin A sin B. (watch the minus sign here! cos P - cos Q = -sin ½(P + Q sin ½(P - Q. Example (7: Express cos 3x in terms of cos x. cos 3x = cos (x + x = cos x cos x sin x sin x = ( cos x - (cos x ( sin x cos x(sin x (Use double angle formulae = ( cos 3 x cos x ( sin x cos x = ( cos 3 x cos x ( ( - cos x(cos x (Use cos x +sin x = = ( cos 3 x cos x ( cos x - cos 3 x = 4 cos 3 x 3 cos x. Note: The corresponding working for sin 3x is similar. sin 3x = sin (x + x = sin x cos x + cos x sin x = ( sin x cos x(cos x + ( sin x(sin x (Use double angle formulae = ( sin x cos x + (sin x - sin 3 x = ( (sin x( - sin x + (sin x - sin 3 x (Use cos x +sin x = = ( sin x sin 3 x + (sin x - sin 3 x = 3 sin x - 4 sin 3 x.

9 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 9 of 7 Example (8: Prove that sin tan cos sec. (Copyright OUP, Understanding Pure Mathematics, Sadler & Thorning, ISBN , Exercise 4D, Q.3a sin tan cos sec sin sin cos cos sec sin cos sec cos Multiplying both sides by cos we have the standard identity sin cos, because sec and cos are reciprocals of each other. Example (9: Prove that sin cos csc. cos sin (The software on the computer uses csc for cosec. (Copyright OUP, Understanding Pure Mathematics, Sadler & Thorning, ISBN , Exercise 4D, Q.8a sin cos csc cos sin sin ( cos ( cos (sin csc sin cos cos ( cos (sin csc cos ( cos (sin csc ( cos ( cos (sin csc csc sin LHS = RHS because cosec and sin are reciprocals of each other. csc cot tan Example (0: Prove that cos. (Copyright OUP, Understanding Pure Mathematics, Sadler & Thorning, ISBN , Exercise 4D, Q.8b cos sin sin cos cos sin sin cos sin cos We begin with cot tan. Therefore sin cos. cot tan csc cot tan cot tan Hence csc (csc (sin cos and finally csc cos cot tan (because cosec and sin are reciprocals of each other.

10 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 0 of 7 Example (: Prove that cos 4 sin 4 cos. (Copyright OUP, Understanding Pure Mathematics, Sadler & Thorning, ISBN , Exercise 4D, Q.8d cos 4 sin 4 cos cos 4 sin 4 cos (cos sin cos sin cos ((cos sin cos cos cos (Both LHS and RHS are expressions for cos. Example (: Prove that sec 4 sec tan 4 tan. (Copyright OUP, Understanding Pure Mathematics, Sadler & Thorning, ISBN , Exercise 4D, Q.8e Since sec tan we havesec 4 sec tan tan tan 4 sec 4 sec tan tan 4 tan = tan 4 tan LHS = RHS. Example (3: Prove the identity: tan( tan 3 60 tan( 60. 3tan (Copyright OCR, GCE Mathematics Paper 473, Jun. 007., Q.9, part (i. We begin with tan( tan tan60 60 tan tan60 tan( tan tan60 60 tan tan60. Multiplying, we have tan tan60 tan tan60 tan( 60 tan( 60 tan tan60 tan tan60 tan tan 60 tan( 60 tan( 60 tan tan 60 (using difference of squares Because tan 60 = we can substitute 3 for tan 60 ; thus tan( tan 3 60 tan( 60 3tan.

11 Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 Solving more trigonometric equations. Identities may be used to re-write an equation in a form that is easier to solve. Example (4 : Solve the equation sin x + cos x - = 0 for - x. We must first use sin x = - cos x to change the equation into a quadratic in cos x. Thus sin x + cos x - = 0 (-cos x + cos x - = 0 - cos x + cos x - = 0 cos x cos x = 0 This factorises at once into (cos x( cos x = 0. cos x = 0 when x = or x = -. cos x =, or cos x = 0.5, when x = 3 or x = -3. The solutions to the equation are therefore x = and 3 (illustrated below. It is important not to simply cancel out cos x from the equation, because there are solutions where cos x = 0, i.e. where x =. (Graph for illustration only students are not expected to sketch it!

12 Mathematics Revision Guides Further Trigonometric Identities and Equations Page of 7 Example (5: Solve the equation cos x + sin x + = 0 for - x. This equation can be turned into a quadratic in sin x by using the identity cos x = - sin x. This gives - sin x + sin x + = 0, or - sin x + sin x + = 0. Using the general formula, we substitute x = sin x, a = -, b = and c =. x b b 4ac a sin 7 4 x. The two solutions are sin x =.8 or sin x = to 3 d.p. The positive value can be rejected at once, since no angle has a sine greater than. The principal value of the other result is c and it falls within the range - x Using the fact that sin (-x = sin x, another possible solution is c, but it falls outside the range. Since the sine function has a period of the other solution in the range is ( or.46 c.

13 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 3 of 7 Example (6: Solve the equation sec = 4 cosec for 0 < <, giving results in radians to 3 decimal places. 4 sin sec = 4 cosec 4 cos sin cos tan 4. The principal solution is =.36 c, but because the tangent function has a period of, another solution is (.36 + c or c. Example (7: Given that sin 3 = 3 sin - 4 sin 3 (see Example 6, use this result to solve, for 0 < < 90, the equation 3 sin 6 cosec = 4. (Copyright OCR, GCE Mathematics Paper 473, Jan. 006., Q.9 part iii. Firstly we substitute =to obtain3 sin 3 cosec = 4, for 0 < < 80 3sin 3 3 sin 3 cosec = 4 4 sin (cosec is reciprocal of sin 3 3(3sin 4sin 4 3(3-4 sin = 4 sin = 4 sin = 0 sin 5 sin 5 sin, or 40., (There is no need to consider the negative value since sin is positive for 0 < < 80 Finally, we must remember that =so we must divide by to give = 0., Example (8: Use the identity tan( tan 3 60 tan( 60 from Example (3, to solve, 3tan for 0 < < 80, the equation tan( + 60 tan(60 = 4sec 3. (Copyright OCR, GCE Mathematics Paper 473, Jun. 007., Q.9, part (ii. tan 3 4sec 3 3tan tan 3 4(tan 3 3 tan (using sec = + tan tan 3 4 tan 3tan tan 3 (4 tan ( 3tan 4 tan 3 4 tan tan 3tan 4 tan 3tan 4 tan tan 3 0 (bringing RHS over to left tan tan 4 0 tan 4 tan 4 = tan - ( 3 3 = 37., The negative value is outside the range, but must be included since the tangent function repeats every 80, and ( or 4.8 is also within the range. Solutions are = 37., 4.8.

14 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 4 of 7 Solution of trigonometric equations of the form a cos b sin = c. (Harmonic Form Equations of the form above can be rewritten in the form R cos (or R sin (for suitable choices of angles and The four compound angle identities are : R cos( + = R cos cos - R sin sin R cos( - = R cos cos + R sin sin R sin(+ = R sin cos + R cos sin R sin( - = R sin cos - R cos sin Any expression of the form a cos b sin can be rewritten to match the above forms. Example (9: Express 3 cos 4 sinin the form R cos (Give in degrees, and from there solve 3 cos 4 sin.5 for R cos( - = 3 cos 4 sin is the closest match : R cos cos R sin sin 3 cos 4 sin Equating sine and cosine terms we have: R cos cos 3 cos Rcos 3. R sin sin sin R sin 4. Squaring the right-hand sides gives R cos = 9 and R sin = 6 R (cos + sin = 5. The bracketed expression is simply, therefore R = When any expression of the form a cos b sin is rewritten in the form R cos (or R sin (then R = a b. To find we see that R sin 4, or that tan = R cos 3 3 cos 4 sin5 cos (53. To solve 5 cos (53..5 for 0 360, first divide by 5 to give cos ( for Two values of (53. satisfying the condition are 60 and 60 ; 3. or 6.9. The second value is out of the required range for but by adding 360 to it we obtain 353., which is within the range. 4, and thus

15 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 5 of 7 Example (0: Express sin - cosin the form R sin (Give in radians. R sin( - = sin - cos is the closest match : R sin cos - R cos sin sin - cos Equating sine and cosine terms we have: R sin cos sin R cos =. R cos sin cos R sin. Hence R = 5 and tan =, and thus.07 c. sin - cos5 sin (.07 c Example (: Express 5 sin cosin the form R sin (Give in radians. R sin( + = 5 sin cos is the closest match. R sin cos R cos sin 5 sin cos R cos 5; R sin R = 5 3 and tan 5 c 5 sin cos3 sin (.76 c Example (: The water temperature ( C of a swimming pool is modelled on the equation W = 7 - sin (5t -. cos(5t, where t is the time in hours after midnight. (The multiple of 5 enters into the questions because the period of 360 degrees needs modifying into a period of 4 hours: 4 5 = 360. i Express the model equation in the form W = 7 - R sin (5t +, where is positive. ii Show that the maximum temperature of the water cannot exceed 30 C according to the model. iii Find the time of day when the water reaches this maximum temperature. iv The management decide that the water temperature should not fall below 6 C during the opening hours of between 0730 and 00. Does the given model satisfy the requirements? i We must rearrange the brackets in the formula first: W = 7 - sin (5t -. cos(5t W = 7 - ( sin (5t +. cos(5t Then continuing in the usual way we have R sin(5t + = sin(5t. cos(5t. R sin(5t cos R cos(5t sin sin(5t. cos(5t. Equating sine and cosine terms we have: R sin(5t cos sin(5t R cos =. R cos(5t sin. cos(5t R sin.. Hence R = ( ( or.973, and tan = the model equation is given by W = sin (5t =., hence ii Since the sine function cannot values outside the range - to, the temperature of the water cannot exceed ( C, or 9.97 C according to this model.

16 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 6 of 7 iii This maximum temperature is reached when sin (5t = - (remember, we re subtracting!, i.e. when 5t = 70 5t =.3 t = 4.8 hours after midnight, or at about 449. iv (A sketch graph is useful here. The temperature of the water is at a maximum at about 500, and so it must take a minimum at about 0300, because this is a sine function with a 4-hr period. By sketching, it can be seen that the water temperature exceeds 6 from about 0800 to 00. We must solve the equation sin (5t = sin (5t = sin (5t = sin (5t = (5t = 9.654, t = -8.07,.6 The answers in degrees must finally be converted into hours after midnight by first adding 360 to the negative value, giving 33.93, and then dividing both results by 5. Hence the water temperature = 6 C when t =.3 hours after midnight, or 08, or when t = 7.5 hours after midnight, or 073. By looking at the graph, the water temperature is above 6 between 073 and 08, which practically coincide with the opening hours!

17 Mathematics Revision Guides Further Trigonometric Identities and Equations Page 7 of 7 The angle between two straight lines. Example (3: Two straight lines have the equations x + y 5 = 0 and 3x y - = 0 respectively. What is the acute angle between them? Each of the lines makes a particular angle with the x-axis, measured anticlockwise. Thus is the angle between the line x + y 5 = 0 and the x axis, while is the angle between the line 3x - y - = 0 and the x axis. The gradient of the line x + y - 5 = 0 is also the tangent of, and its value is -½. Similarly the gradient of the line 3x y - = 0 is the tangent of, namely 3. The acute angle between the lines is, and, by the triangle laws, = -. Taking tangents we have tan = tan ( - or tan tan tan tan tan Substituting tan = -½ and tan = 3, we can calculate the acute angle between the two lines using ( 3 3 tan 7 = 8.9 or.49 c. ( (3 Also note that the constant terms in the equations of the lines do not affect the result. Thus the acute angle between the lines x + y - 6 = 0 and 3x y + 5 = 0 would also be 8.9 or.49 c. The general formula for the angle between two straight lines (not in scope of syllabus, given their gradients m and m, is tan m m m m. (It is not important which gradient is chosen as m ; if tan comes out negative, then disregarding the sign will still give the acute angle result.

### SOLVING TRIGONOMETRIC EQUATIONS

Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC

### TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.

Pre-Calculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle

### Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

### Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring

Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest

### 1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### Trigonometric Functions: The Unit Circle

Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry

### Core Maths C3. Revision Notes

Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

### 4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles

4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred

### Right Triangle Trigonometry

Section 6.4 OBJECTIVE : Right Triangle Trigonometry Understanding the Right Triangle Definitions of the Trigonometric Functions otenuse osite side otenuse acent side acent side osite side We will be concerned

### Section 6-3 Double-Angle and Half-Angle Identities

6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

### Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

### y = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions

MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### ALGEBRA 2/ TRIGONOMETRY

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA 2/ TRIGONOMETRY Friday, June 14, 2013 1:15 4:15 p.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 28.......................

### National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

### Graphing Trigonometric Skills

Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### Trigonometry Review Workshop 1

Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions

### Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 42. The sum of the measures of the angles of a triangle is 180. Therefore, The sine of an angle

### Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Trigonometric Identities and Equations

LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now

### 4.1 Radian and Degree Measure

Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### RIGHT TRIANGLE TRIGONOMETRY

RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will

### Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

### y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Mathematics Higher Tier, Algebraic Fractions

These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton

### D.3. Angles and Degree Measure. Review of Trigonometric Functions

APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

### 5.2 Unit Circle: Sine and Cosine Functions

Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### opp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles

Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

### Chapter 6: Periodic Functions

Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values

### Angles and Their Measure

Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

### 6.1 Basic Right Triangle Trigonometry

6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch

### Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC4. (Specification 6360) Pure Core 4. Final.

Version.0 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Sample Problems. 10. 1 2 cos 2 x = tan2 x 1. 11. tan 2 = csc 2 tan 2 1. 12. sec x + tan x = cos x 13. 14. sin 4 x cos 4 x = 1 2 cos 2 x

Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### Semester 2, Unit 4: Activity 21

Resources: SpringBoard- PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities

### Section 5-9 Inverse Trigonometric Functions

46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

### Trigonometry Chapter 3 Lecture Notes

Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece

### a cos x + b sin x = R cos(x α)

a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this

5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### Unit 6 Trigonometric Identities, Equations, and Applications

Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Section 6-4 Product Sum and Sum Product Identities

480 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Section 6-4 Product Sum and Sum Product Identities Product Sum Identities Sum Product Identities Our work with identities is concluded by developing

### 1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic

1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single

### Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right

### Evaluating trigonometric functions

MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,

### STRAND: ALGEBRA Unit 3 Solving Equations

CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

### 1. Introduction circular deﬁnition Remark 1 inverse trigonometric functions

1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric

### Notes and questions to aid A-level Mathematics revision

Notes and questions to aid A-level Mathematics revision Robert Bowles University College London October 4, 5 Introduction Introduction There are some students who find the first year s study at UCL and

### Edexcel AS and A-level Modular Mathematics

Core Mathematics Edexcel AS and A-level Modular Mathematics Greg Attwood Alistair Macpherson Bronwen Moran Joe Petran Keith Pledger Geoff Staley Dave Wilkins Contents The highlighted sections will help

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Geometry Mathematics Curriculum Guide Unit 6 Trig & Spec. Right Triangles 2016 2017

Unit 6: Trigonometry and Special Right Time Frame: 14 Days Primary Focus This topic extends the idea of triangle similarity to indirect measurements. Students develop properties of special right triangles,

### TRANSFORMATIONS OF GRAPHS

Mathematics Revision Guides Transformations of Graphs Page 1 of 24 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C1 Edexcel: C1 OCR: C1 OCR MEI: C1 TRANSFORMATIONS OF GRAPHS Version

### Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles

Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### INVERSE TRIGONOMETRIC FUNCTIONS

Mathematics, in general, is fundamentally the science of self-evident things. FELIX KLEIN. Introduction In Chapter, we have studied that the inverse of a function f, denoted by f, eists if f is one-one

MTN Learn Mathematics Grade 10 radio support notes Contents INTRODUCTION... GETTING THE MOST FROM MINDSET LEARN XTRA RADIO REVISION... 3 BROADAST SCHEDULE... 4 ALGEBRAIC EXPRESSIONS... 5 EXPONENTS... 9

### GCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.

GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

### Right Triangles and SOHCAHTOA: Finding the Measure of an Angle Given any Two Sides (ONLY for ACUTE TRIANGLES Why?)

Name Period Date Right Triangles and SOHCAHTOA: Finding the Measure of an Angle Given any Two Sides (ONLY for ACUTE TRIANGLES Why?) Preliminary Information: SOH CAH TOA is an acronym to represent the following

### Core Maths C2. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### Trigonometry LESSON ONE - Degrees and Radians Lesson Notes

210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:

### Version. General Certificate of Education (A-level) January 2013. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version General Certificate of Education (A-level) January Mathematics MPC (Specification 66) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together with

### 2015 Junior Certificate Higher Level Official Sample Paper 1

2015 Junior Certificate Higher Level Official Sample Paper 1 Question 1 (Suggested maximum time: 5 minutes) The sets U, P, Q, and R are shown in the Venn diagram below. (a) Use the Venn diagram to list

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

### Introduction Assignment

PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying

### MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

### Chapter 6 Trigonometric Functions of Angles

6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.

### how to use dual base log log slide rules

how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102

### y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C

Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

### Basic Electrical Theory

Basic Electrical Theory Mathematics Review PJM State & Member Training Dept. Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different

### x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3

CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### 2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

### You can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure

Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve

### Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.

### Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.

Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where

### Integration Involving Trigonometric Functions and Trigonometric Substitution

Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving

Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

### 2015 Mathematics. New Higher Paper 1. Finalised Marking Instructions

National Qualifications 0 0 Mathematics New Higher Paper Finalised Marking Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications