# Trigonometry Hard Problems

Save this PDF as:

Size: px
Start display at page:

Download "Trigonometry Hard Problems"

## Transcription

1 Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory. The problem does not say the lake is a circle, but if it is not, the problem cannot be solved. So, let s assume the lake is a circle. What is a transit? From we get The transit is used by the surveyor to measure both horizontal and vertical angles. A transit A piling What is a piling? From wisegeek.com, a piling is a component in a foundation which is driven into the ground to ensure that the foundation is deep. We will need a right angle to solve a problem with only one length and angle given, so let s infer a 90 angle as shown in the following diagram. Based on the diagram, which meets every criterion laid out in the problem, we now have: tan tan 35 ft. The body of water in this problem would be better described as a pond than a lake! Page 1 of 10

2 Based on the illustration at right, we get the following: tan tan.45 The key on this type of problem is to draw the correct triangle. Recall that tan Since is in 2, we can plot the point 3, 8 to help us. Notice that the hypotenuse must have length: Then, cos The angle is in Q2, but tangent is defined only in Q1 and Q4. Further, tan 0 in Q2. So we seek the angle in Q4, where tangent is also 0, with the same tangent value as. Recall that the tangent function has a period of radians. Then, tan tan Page 2 of 10

3 5) Find the exact value of the expression. a. cos b. sin c. tan Know where the primary values for the inverse trig functions are defined. sin θ is defined in Q1 and Q4. cos θ is defined in Q1 and Q2. tan θ is defined in Q1 and Q4. You can think of this as adding two vectors with the bearings identified in the problem. Step 1: Convert each vector into its i and j components, using reference angles. Let u be a vector of 25 mi. at bearing: N63 W From the diagram at right, θ cos sin Let v be a vector of 24 mi. at bearing: S27 W From the diagram at right, φ cos sin Step 2: Add the results for the two vectors , , , Step 3: Find the resulting angle and convert it to its bearing form (see diagram above right). tan Bearing S W. Page 3 of 10

4 7) A ship is 50 miles west and 31 miles south of a harbor. What bearing should the Captain set to sail directly to harbor? θ tan 31.8 φ For Problems 8 and 9, use: or with 0 Harmonic motion equations: cos or sin The spring will start at a value of 8, and oscillate from 8 to 8 over time. A good representation of this would be a cosine curve with lead coefficient 8, because cos0 1, whereas, sin0 0. The period of our function is 5 seconds. So, we get: 1 period The resulting equation, then, is: Note: both of these functions can be graphed using the Trigonometry app available at Assuming that 0 at time 0, it makes sense to use a sine function for this problem. Since the amplitude is 3 feet, a good representation of this would be a sine curve with lead coefficient 3, because sin0 0, whereas, cos0 0. Recalling that 2, and with we get: 2 5. The resulting equation, then, is: Page 4 of 10

5 cos sin 1 tan cos sin cos cos sin cos cos sin cos sin cos cos sin 1 cos cos sin sin sin cot sin sin cos sin sin cos Answer B sin 75 sin cos Note: the half angle formula has a "" sign in front. We use for this problem because 75 is in Q1, and sine values are positive in Q1. cos 3 8 cos cos Note: the half angle formula has a "" sign in front. We use for this problem because is in Q1, and cosine values are positive in Q1. Page 5 of 10

6 tan sintan tan sin tan 0 tan sin 1 0 tan 0 or sin 1 0 0,π sin 1, While is a solution to the equation sin 1, tan is undefined at. Therefore, the original equation, tan sintan is undefined at. So is not a solution to this equation. 2 cos sin sin sin20 22sin sin20 2 sin sin0 sin 2 sin 1 0 When an equation contains more than one function, try to convert it to one that contains only one function. sin 0 or 2 sin 1 0 0,π sin,,,, Page 6 of 10

7 cos a 1, 1 4, cos cos b 1, 2 3, cos cos Page 7 of 10

8 Substitute cos and Substitute xr cos Substitute cos,sin and Page 8 of 10

9 a) The process of putting a complex number in polar form is very similar to converting a set of rectangular coordinates to polar coordinates. So, if this process seems familiar, that s because it is tan tan 3 in So, the polar form is: 6 cos sin b) tan tan 3 3 in So, the polar form is: 10 cos sin 3 cos 15 sin 15 To take a power of two numbers in polar form, take the power of the value and multiply the angle by the exponent. (This is the essence of DeMoivre s Theorem.) 3 cos 15 sin 15 3 cis cis60 81 cis60 81 cos 60 sin Page 9 of 10

10 tan tan 3 in Then, cis 6 cis cis cos 2 sin 2 Page 10 of 10

### Trigonometry (Chapter 6) Sample Test #1 First, a couple of things to help out:

First, a couple of things to help out: Page 1 of 20 More Formulas (memorize these): Law of Sines: sin sin sin Law of Cosines: 2 cos 2 cos 2 cos Area of a Triangle: 1 2 sin 1 2 sin 1 2 sin 1 2 Solve the

More information

### 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

More information

### Who uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)

1- The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview

More information

### How to Graph Trigonometric Functions

How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

More information

### Mid-Chapter Quiz: Lessons 4-1 through 4-4

Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to

More information

### 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

More information

### Finding trig functions given another or a point (i.e. sin θ = 3 5. Finding trig functions given quadrant and line equation (Problems in 6.

1 Math 3 Final Review Guide This is a summary list of many concepts covered in the different sections and some examples of types of problems that may appear on the Final Exam. The list is not exhaustive,

More information

### 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

More information

### 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

More information

### 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

More information

### TRIGONOMETRY. circular functions. These functions, especially the sine and

TRIGONOMETRY Trigonometry has its origins in the study of triangle measurement. Natural generalizations of the ratios of righttriangle trigonometry give rise to both trigonometric and circular functions.

More information

### 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

More information

### Section 3.1 Radian Measure

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

More information

### 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

More information

### 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

More information

### Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain

More information

### 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

More information

### Unit 8 Inverse Trig & Polar Form of Complex Nums.

HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric

More information

### 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.

More information

### 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

More information

### 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

More information

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

### 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:

More information

### 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.

More information

### (1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 2. Right Triangle Trigonometry

2. Right Triangle Trigonometry 2.1 Definition II: Right Triangle Trigonometry 2.2 Calculators and Trigonometric Functions of an Acute Angle 2.3 Solving Right Triangles 2.4 Applications 2.5 Vectors: A Geometric

More information

### 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

More information

### 6.3 Polar Coordinates

6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

More information

### 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,

More information

### Algebra 2/ Trigonometry Extended Scope and Sequence (revised )

Algebra 2/ Trigonometry Extended Scope and Sequence (revised 2012 2013) Unit 1: Operations with Radicals and Complex Numbers 9 days 1. Operations with radicals (p.88, 94, 98, 101) a. Simplifying radicals

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### We d like to explore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( x)

Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can

More information

### 4-1 Right Triangle Trigonometry

Find the exact values of the six trigonometric functions of θ. 1. The length of the side opposite θ is 8 is 18., the length of the side adjacent to θ is 14, and the length of the hypotenuse 3. The length

More information

### Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks

Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions

More information

### Unit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6)

Unit 6 Introduction to Trigonometry Inverse Trig Functions (Unit 6.6) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When you have completed this lesson you will: Understand

More information

### 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

More information

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

More information

### 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

More information

### 4-6 Inverse Trigonometric Functions

Find the exact value of each expression, if it exists. 1. sin 1 0 Find a point on the unit circle on the interval with a y-coordinate of 0. 3. arcsin Find a point on the unit circle on the interval with

More information

### 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

More information

### Section 0.4 Inverse Trigonometric Functions

Section 0.4 Inverse Trigonometric Functions Short Recall from Trigonometry Definition: A function f is periodic of period T if f(x + T ) = f(x) for all x such that x and x+t are in the domain of f. The

More information

### AAT Unit 5 Graphing Inverse Trig Functions, Trig Equations and Harmonic Motion. Name: Block: Section Topic Assignment Date Due: 5-7 AAT- 16,19

1 AAT Unit 5 Graphing Inverse Trig Functions, Trig Equations and Harmonic Motion Name: Block: Section Topic Assignment Date Due: 5-7 AAT- 16,19 Inverse Trig Functions Page 566:10-18 even, 36-56 even Page

More information

### 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

More information

### Example #1: f(x) = x 2. Sketch the graph of f(x) and determine if it passes VLT and HLT. Is the inverse of f(x) a function?

Unit 3 Eploring Inverse trig. functions Standards: F.BF. Find inverse functions. F.BF.d (+) Produce an invertible function from a non invertible function by restricting the domain. F.TF.6 (+) Understand

More information

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

More information

### PRE-CALCULUS GRADE 12

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.

More information

### 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

More information

### Inverse Trig Functions

MATH 7 Inverse Trig Functions Dr. Neal, WKU A function y = f (x) is one to one if it is always the case that different x values are assigned to different y values. For example, y = x + 4 is one to one,

More information

### 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.

More information

### 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

More information

### 2312 test 2 Fall 2010 Form B

2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function

More information

### 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

More information

### 6.3 Inverse Trigonometric Functions

Chapter 6 Periodic Functions 863 6.3 Inverse Trigonometric Functions In this section, you will: Learning Objectives 6.3.1 Understand and use the inverse sine, cosine, and tangent functions. 6.3. Find the

More information

### MATH122 Final Exam Review If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71.

MATH1 Final Exam Review 5. 1. If c represents the length of the hypotenuse, solve the right triangle with a = 6 and B = 71. 5.1. If 8 secθ = and θ is an acute angle, find sinθ exactly. 3 5.1 3a. Convert

More information

### TRIGONOMETRY GRADES THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618

TRIGONOMETRY GRADES 11-12 THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618 Board Approval Date: October 29, 2012 Michael Nitti Written by: EHS Mathematics Department Superintendent In accordance

More information

### 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

More information

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

More information

### 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,

More information

### 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

More information

### MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

More information

### 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

More information

### 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

More information

### ALGEBRA 2/TRIGONOMETRY

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession

More information

### 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

More information

### Plane Trigonometry - Fall 1996 Test File

Plane Trigonometry - Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points.

More information

### POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

More information

### WEEK #5: Trig Functions, Optimization

WEEK #5: Trig Functions, Optimization Goals: Trigonometric functions and their derivatives Optimization Textbook reading for Week #5: Read Sections 1.8, 2.10, 3.3 Trigonometric Functions From Section 1.8

More information

### General Physics 1. Class Goals

General Physics 1 Class Goals Develop problem solving skills Learn the basic concepts of mechanics and learn how to apply these concepts to solve problems Build on your understanding of how the world works

More information

### Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates

Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in

More information

### Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time.

Plotting Polar Curves We continue to study the plotting of polar curves. Recall the family of cardioids shown last time. r = 1 cos(θ) r = 1 + cos(θ) r = 1 + sin(θ) r = 1 sin(θ) Now let us look at a similar

More information

### Friday, January 29, 2016 9:15 a.m. to 12:15 p.m., only

ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession

More information

### Trigonometry. The inverse trig functions

Trigonometry The inverse trig functions If you re trying to answer the question, What angle has a sine (or cosine, or tangent) equal to a given value? - say sin x =.3456 you need an inverse operation to

More information

### 4-1 Right Triangle Trigonometry

Find the measure of angle θ. Round to the nearest degree, if necessary. 31. Because the lengths of the sides opposite θ and the hypotenuse are given, the sine function can be used to find θ. 35. Because

More information

### 3. Right Triangle Trigonometry

. Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle

More information

### 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:

More information

### Higher. Functions and Graphs. Functions and Graphs 18

hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

More information

### 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

More information

### 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

More information

### 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,

More information

### Graphs of Polar Equations

Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate

More information

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

### Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

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

More information

### Inverse Trigonometric Functions

Inverse Trigonometric Functions I. Four Facts About Functions and Their Inverse Functions:. A function must be one-to-one (an horizontal line intersects it at most once) in order to have an inverse function..

More information

### 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

More information

### 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

More information

### Any two right triangles, with one other angle congruent, are similar by AA Similarity. This means that their side lengths are.

Lesson 1 Trigonometric Functions 1. I CAN state the trig ratios of a right triangle 2. I CAN explain why any right triangle yields the same trig values 3. I CAN explain the relationship of sine and cosine

More information

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

More information

### 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

More information

### MTH 122 Plane Trigonometry Fall 2015 Test 1

MTH 122 Plane Trigonometry Fall 2015 Test 1 Name Write your solutions in a clear and precise manner. Answer all questions. 1. (10 pts) a). Convert 44 19 32 to degrees and round to 4 decimal places. b).

More information