1. Introduction circular deﬁnition Remark 1 inverse trigonometric functions

Save this PDF as:

Size: px
Start display at page:

Transcription

1

2

3 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 functions. The next section (The triangular definition) of this lesson presents the right triangular definitions of the trigonometric functions. This raises a question about the consistency or agreement of the circular and right triangle definitions. For example,does the sine function produce the same value for a given angle regardless of the definition used. Predictably,the answer is affirmative as verified in section 3. The next two sections (section 4 and section 5) demonstrate how the trigonometric functions can be applied to arbitrary triangles. In Section 6 the circular definition of the trigonometric functions is extended to circles of arbitrary radii. Remark 1 Some examples and exercises in this lesson require the use of a calculator equipped with the inverse trigonometric functions. The inverse sine, cosine and tangent function keys are usually denoted on keypads by sin 1, cos 1, and tan 1. These functions are used to determine the measure of an angle α if sin α (or cos α or tan α) is known. For example, if sin α =.2, then α = sin 1.2= Before executing these functions the reader should ensure that the calculator is in the correct mode. Degree mode is used throughout this lesson. It should be noted that the general theory of inverse functions is rather sophisticated and presently lies beyond the scope of this tutorial.

4 2. The triangular definition Consider the right triangle in Figure 3.1 where α denotes one of the two non-right angles. The side of the triangle opposite the right angle is called the hypotenuse. The remaining two sides of the triangle can be uniquely identified by relating D O F JA K I A = =? A JI A Figure 3.1: triangle F F I EJA I A Right them to the angle α as follows. The adjacent side (or the side adjacent α) refers to the side that,along with the the hypotenuse,forms the angle α. The third side of the triangle is called the opposite side (or the side opposite α). The dependence of these labels on α is crucial since,for example, the side opposite α is adjacent to the other non-right angle of the triangle. The abbreviations hyp for the length of the hypotenuse,and opp and adj for the lengths of the opposite and adjacent sides respectively are used to define the values of the six trigonometric functions for the angle α 90. These definitions are given in Table 3.1. sin α = opp hyp cot α = adj opp adj cos α = hyp hyp sec α = adj opp tan α = adj csc α = hyp opp Table 3.1: Definition of the trigonometric functions from a right triangle.

5 Section 2: The triangular definition 5 Example 2 A right triangle with angle α 90 has an adjacent side 4 units long and a hypotenuse 5 units long. Determine sin α and tan α. Also, determine sin β and tan β where β denotes the second non-right angle of the triangle. Finally, use a calculator to determine α and β. Solution: By the Pythagorean Theorem =4 2 +(opp) 2, so the length of the side opposite α is 3= units. Consequently, sin α = opp hyp = 3 5 and Using a calculator (in degree mode) we find α = sin opp tan α = adj = 3 4. For the angle β, the roles of adjacent and opposite sides must be reversed. Hence, sin β = 4 5 and tan β = 4 3. Again, using a calculator we find β = sin 1 4/ A sketch of the triangle is given below. 1 The lengths of the sides of a triangle satisfy (adj) 2 +(opp) 2 =(hyp) 2.

6 Section 2: The triangular definition 6! \$ & % = # " > #!! The reader should observe that =90.0 which serves as a partial confirmation, 2 but not a guarantee, of the correctness of the above calculations.! 2 The values of the three interior angles of a triangle sum to 180. Hence, the two non-right angles of a right triangle must sum to 90.

7 Section 2: The triangular definition 7 Example 3 A right triangle with angle α =30 has an adjacent side 4 units long. Determine the lengths of the hypotenuse and side opposite α. Solution: The definition cos α = adj hyp suggests that cos 30 = 4 hyp. So, Similarly, sin 30 = opp hyp hyp = 4 cos 30 = 4 3/2 = 8 3. so the length of the side opposite α is & ( )( ) 8 1 opp = hyp sin 30 = 3 = 4.! " 2 3!! " The Pythagorean Theorem provides a quick endorsement of the computed values. Specifically, note that ( ) = ( 3 = ) = 64 ( ) = 3.

8 3. Consistency of the triangular definition As illustrated in Figure 3.2,several right triangles may contain the same angle α. Triangles with the same angles but different side lengths are called similar. Similar triangles,then,have the same shape and differ only in size. This raises an immediate concern about using the definitions in Table 3.1. Specifically,do the values sin α, cos α,and the remaining trigonometric functions change with the size of the triangle? The answer is no as verified = by the following argument. Since the two right triangles = in Figure 3.2 are similar,geometric considerations ensure that the ratios of corresponding sides of the triangles satisfy ) O Figure 3.2: Similar right H = o h, A H = a h, and o a = O A. triangles. These equalities and the definitions in Table 3.1 suggest that the values of the trigonometric functions for the angle α are independent of the the triangle used to obtain them. For example sin α = O H = o h. D 0

9 Section 3: Consistency of the triangular definition 9 Also,using the larger triangle we have tan α = O/H A/H = O A, while the smaller triangle suggests tan α = o/h a/h = o a. Since o = O, tan α remains unchanged as the size,but not the shape,of the triangle a A fluctuates. Similar reasoning verifies the consistency of the triangle definitions of all the trigonometric functions.

10 Section 3: Consistency of the triangular definition 10 Example 4 Consider the right triangle in Figure 3.3. Find the length of the side adjacent to α if sin α =3/5. = % Figure 3.3: A right triangle. Solution: There is no way to compute the length of the adjacent side directly so we first compute the length of the hypotenuse. From Figure 3.3and the triangular definition of the sine function we have sin α = opp hyp = 7 hyp. Since sin α = 3 5 we have 7 hyp = 3 5 = hyp = 35 3.

11 Section 3: Consistency of the triangular definition 11 The Pythagorean Theorem can now be applied to determine the adjacent side as follows (35 ) adj = (7) 3 2 = 9 = Of course the fundamental properties of similar triangles could have been used to solve this problem. Because sin α =3/5, the given triangle in Figure 3.3 is similar to the triangle with angle α, a hypotenuse of length 5, and side opposite α of length 3. The similarity of these triangles is illustrated in the figure below. The Pythagorean Theorem indicates that the side opposite α in the smaller triangle has length 25 9=4. # =! " Because of this similarity we have the equality adj 7 = 4 3. Solving this equation gives adj = = adj %

12 Section 3: Consistency of the triangular definition 12 Example 5 Determine the angles β and γ and the lengths of the sides a and c in the triangle Figure (a) below. Solution: Construct the line segment h that is perpendicular to b connecting the angle β to the side b as indicated in Figure (b). Doing so forms two right triangles T 1 and T 2 with a common side h and base sides b 1 and b 2. Since sin 30 = h/4 we have h = 4 sin 30 =2. Also, ( ) ( ) h 2 sin γ = 2 = 2 2 = 1, 2 2 so γ =45. Since β = 180, we have β = = 105. Finally, b = b 1 + b 2 = 2 cot cot 45 =2( 3+1)= ! " > b (a) C! " b 1 h T 1 T 2 (b) b 2 C

13 4. Law of Sines Example 5 is suggestive of a general rule called the Law of Sines. Specifically,given the sample triangle in Figure 3.4 with sides a, b, and c c > a opposite the angles α, β, and γ respectively,the = C Law of Sines states that b sin α = sin β = sin γ. (1) a b c Figure 3.4: Sample triangle. To prove this consider the triangle in the figure below in which a line segment of length h is constructed perpendicular to side b connecting b to the angle β. c > a h = C b The two right triangles thus formed suggest that h = c sin α and h = a sin γ.

14 Section 4: Law of Sines 14 Hence, c sin α = a sin γ = sin α = sin γ. (2) a c A similar argument using the triangle below verifies that sin (180 β) b = sin γ. c h c > a = C b Since sin β = sin (180 β) we have sin β = sin γ. b c Combining this equality with Equation 2 establishes Equation 1.

15 Section 4: Law of Sines 15 Example 6 Reconsider Example 5. (The figure for that example is reproduced below.) The Law of Sines (Equation 1) facilitates the calculation of the remaining parts of the triangle since sin = sin γ 4 so that sin γ = 4(1/2) 2 2 = 1. 2 Hence, γ =45. Also, β = = 105 so that sin 30 2 sin 105 = = b =4 2 sin 105 = b A calculator was used to compute sin 105.! " b > C

16 Section 4: Law of Sines 16 Example 7 Actually, the conditions placed on the triangle in Example 6 permit two solutions when using the Law of Sines. Figure (a) below indicates that the angle γ =45 in that example could be replaced with the angle γ = (180 γ) = = 135. # " "! C b C! " \$ "! # (a) (b) In this case β = 180 ( )=15. Hence, b =4 2 sin 15 = The second solution is depicted in Figure (b).

17 As the previous example illustrates,the Law of Sines does not always have a unique solution. Specifically,it is possible that two triangles possess a given angle and specified sides adjacent and opposite that angle. This is called the ambiguous case of the Law of Sines. There are rules for determining when the ambiguous case produces no solution,a unique solution,or two solutions. However,perhaps the best way of determining this is to simply solve the problem for as many solutions as possible. This strategy is illustrated in the exercises. 5. Law of Cosines A second law that deserves attention is the Law of Cosines which is presented without justification. For a triangle with sides a, b, and c opposite the angles α, β, and γ respectively,the Law of Cosines states that a 2 = b 2 + c 2 2bc cos α. (3) Observe that this rule reduces to the Pythagorean Theorem if α is a right angle since cos 90 = 0. This law is valid for any of the three angles of the triangle so it could have been stated as b 2 = a 2 + c 2 2ac cos β or c 2 = a 2 + b 2 2ab cos γ.

18 Section 5: Law of Cosines 18 Example 8 Suppose a triangle has adjacent sides of lengths 2 and 3 with an interior angle of measure α =70. (See the figure below.) Then by the Law of Cosines the length of the side a opposite the angle α is given by a = (2)(3) cos 70 = = = > = %! C Observe that the Law of Cosines can be used to find β. Indeed, cos β = 1 b 2 + a 2 + c 2 2 ac = ( ) (2) =

19 Section 5: Law of Cosines 19 so that β = cos = Likewise, since c 2 = a 2 + b 2 2ab cos γ, we see that cos (γ) = Hence, γ = cos = As a check note that the sum of these three angles is Of course, the Law of Sines could also have been used to determine the measure of β and γ. For example, since sin α = sin β a b we have sin = sin β 3 = sin β = 3 sin = Hence, β = sin 1 ( ) = Note that this last answer is not exactly the same as that obtained for β using the Law of Cosines. This demonstrates some of the difficulties with numerical calculations.

20 6. The circular definition revisited The observation that the values of trigonometric functions are independent of the size of the right triangle suggests that the definition given for these functions on the unit circle can be modified to include circles of arbitrary radii. Examination of the figure below indicates that the values of the functions are those given in the table. The sides of the right triangle in the figure satisfy adj = x, opp = y, and hyp = r. Note that if the circle is the unit circle so that r = 1,then these values reduce to those given in Table 2.1 in Lesson 2. O N O N O H H J N O N sin t = y = opp,cos t = x = adj, r hyp r hyp tan t = y = opp,cot t = x = adj, x adj y opp sec t = r y = hyp adj,csc t = r x = hyp opp.

21 7. Exercises Exercise 1. A right triangle contains a 35 angle that has an adjacent side of length 4.5 units. How long is the opposite side? How long is the hypotenuse? (You will need a calculator for this problem. Remember to set it to degree mode.) Exercise 2. Suppose sin α =4/7. Without using a calculator find cos α and tan α. Exercise 3. Let α denote a non-right angle of a right triangle. Prove that sin α = cos (90 α). Observe that a similar identity holds for the other five trigonometric functions. Exercise 4. Determine the length of the chord PQ in the figure below. O 2 N O " F! N 3

22 Section 7: Exercises 22 Exercise 5. Let T be a triangle with sides a, b, and c opposite the angles α, β, and γ respectively as depicted in the figure below.? > = = > In each problem below determine the remaining parts of T if such a triangle exists. Remember that some conditions may permit two solutions. (See Example 7.) (a) a =10,b=7, and α =80 (b) a =5,b=7, and α =40 (c) a =5,b=7, and c =10 C

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

11 Trigonometric Functions of Acute Angles

Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common

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

Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge

Section 9.4 Trigonometric Functions of any Angle

Section 9. Trigonometric Functions of any Angle So far we have only really looked at trigonometric functions of acute (less than 90º) angles. We would like to be able to find the trigonometric functions

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

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

Student Academic Learning Services Page 1 of 6 Trigonometry

Student Academic Learning Services Page 1 of 6 Trigonometry Purpose Trigonometry is used to understand the dimensions of triangles. Using the functions and ratios of trigonometry, the lengths and angles

Basic Trigonometry, Significant Figures, and Rounding - A Quick Review

Basic Trigonometry, Significant Figures, and Rounding - A Quick Review (Free of Charge and Not for Credit) by Professor Patrick L. Glon, P.E. Basic Trigonometry, Significant Figures, and Rounding A Quick

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

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

1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

Trigonometry I. MathsStart. Topic 5

MathsStart (NOTE Feb 0: This is the old version of MathsStart. New books will be created during 0 and 04) Topic 5 Trigonometry I h 0 45 50 m x MATHS LEARNING CENTRE Level, Hub Central, North Terrace Campus

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

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

Trigonometry. An easy way to remember trigonometric properties is:

Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.

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

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

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

Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

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

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

Intermediate Algebra with Trigonometry. J. Avery 4/99 (last revised 11/03)

Intermediate lgebra with Trigonometry J. very 4/99 (last revised 11/0) TOPIC PGE TRIGONOMETRIC FUNCTIONS OF CUTE NGLES.................. SPECIL TRINGLES............................................ 6 FINDING

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

Triangle Trigonometry and Circles

Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of

Types of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs

MTH 065 Class Notes Lecture 18 (4.5 and 4.6) Lesson 4.5: Triangles and the Pythagorean Theorem Types of Triangles Triangles can be classified either by their sides or by their angles. Types of Angles An

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:

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

Similarity, Right Triangles, and Trigonometry

Instruction Goal: To provide opportunities for students to develop concepts and skills related to trigonometric ratios for right triangles and angles of elevation and depression Common Core Standards Congruence

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

Inverse Trigonometric Functions

SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate

FURTHER TRIGONOMETRIC IDENTITIES AND EQUATIONS

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

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

6.6 The Inverse Trigonometric Functions. Outline

6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous

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

Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook

Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.

2.1 The Tangent Ratio

2.1 The Tangent Ratio In this Unit, we will study Right Angled Triangles. Right angled triangles are triangles which contain a right angle which measures 90 (the little box in the corner means that angle

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

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

Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY

CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that

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

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS

Basic numerical skills: TRIANGLES AND TRIGONOMETRIC FUNCTIONS 1. Introduction and the three basic trigonometric functions (simple) Triangles are basic geometric shapes comprising three straight lines,

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

Unit 3 Right Triangle Trigonometry - Classwork

Unit 3 Right Triangle Trigonometry - Classwork We have spent time learning the definitions of trig functions and finding the trig functions of both quadrant and special angles. But what about other angles?

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

4.4 Right Triangle Trigonometry

4.4 Right Triangle Trigonometry Trigonometry is introduced to students in two different forms, as functions on the unit circle and as functions on a right triangle. The unit circle approach is the most

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

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

Trigonometric Identities

Trigonometric Identities Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout dpresents some of the most useful trigonometric identities. It also explains how to derive new

Teaching & Learning Plans. Plan 8: Introduction to Trigonometry. Junior Certificate Syllabus

Teaching & Learning Plans Plan 8: Introduction to Trigonometry Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes

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

Using Trigonometry to Find Missing Sides of Right Triangles

Using Trigonometry to Find Missing Sides of Right Triangles A. Using a Calculator to Compute Trigonometric Ratios 1. Introduction: Find the following trigonometric ratios by using the definitions of sin(),

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

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

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

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

MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions

Section 8.1: The Inverse Sine, Cosine, and Tangent Functions The function y = sin x doesn t pass the horizontal line test, so it doesn t have an inverse for every real number. But if we restrict the function

Analytic Trigonometry

Name Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions and

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

BC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because

150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,

5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

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

HYPERBOLIC TRIGONOMETRY

HYPERBOLIC TRIGONOMETRY STANISLAV JABUKA Consider a hyperbolic triangle ABC with angle measures A) = α, B) = β and C) = γ. The purpose of this note is to develop formulae that allow for a computation of

Triangle Definition of sin and cos

Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ

Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

9.1 Trigonometric Identities

9.1 Trigonometric Identities r y x θ x -y -θ r sin (θ) = y and sin (-θ) = -y r r so, sin (-θ) = - sin (θ) and cos (θ) = x and cos (-θ) = x r r so, cos (-θ) = cos (θ) And, Tan (-θ) = sin (-θ) = - sin (θ)

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:

First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin

7.1 Apply the Pythagorean Theorem

7.1 Apply the Pythagorean Theorem Obj.: Find side lengths in right triangles. Key Vocabulary Pythagorean triple - A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation

Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics.

MATH19730 Part 1 Trigonometry Section2 Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics. An angle can be measured in degrees

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,

Solutions to Exercises, Section 6.1

Instructor s Solutions Manual, Section 6.1 Exercise 1 Solutions to Exercises, Section 6.1 1. Find the area of a triangle that has sides of length 3 and 4, with an angle of 37 between those sides. 3 4 sin

as a fraction and as a decimal to the nearest hundredth.

Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. sin A The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, 2. tan C The tangent of an

alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

29 Wyner PreCalculus Fall 2016

9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

Pythagorean Theorem: 9. x 2 2

Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2

1 Solution of Homework

Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,

Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

Page. Trigonometry Sine Law and Cosine Law. push

Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures.

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.

Geometry in a Nutshell

Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where

Section 8 Inverse Trigonometric Functions

Section 8 Inverse Trigonometric Functions Inverse Sine Function Recall that for every function y = f (x), one may de ne its INVERSE FUNCTION y = f 1 (x) as the unique solution of x = f (y). In other words,

Law of Cosines TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System

Math Objectives Students will be able to state the Law of Cosines Students will be able to apply the Law of Cosines to find missing sides and angles in a triangle Students will understand why the Law of

POTENTIAL REASONS: Definition of Congruence:

Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

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

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

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

Chapter 12. The Straight Line

302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

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