1 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 to measure it directly. For example, surveyors use trigonometry to measure the heights of mountains and distances across bodies of water, while engineers and architects use trigonometry to ensure their buildings are being built to exact specifications. Also, any phenomena that repeats itself on a periodic basis the seasons, fluctuations in sales, heartbeats can be modeled by trigonometric functions. Section 7.1: Introduction to Trigonometry Angles and Quadrants We start with a discussion of angles. A ray is placed with its endpoint at the origin of an xy-axis system, with the ray itself lying along the positive x-axis. The ray is allowed to rotate. Counterclockwise rotations are considered positive, and clockwise rotations are called negative (See Fig 1). The xy-axis system itself is subdivided into four quadrants, each defined by the x- and y-axes. These quadrants are numbered 1 through 4, with Quadrant 1 being at the top right (where both x and y are positive) and Quadrants, 3 and 4 following consecutively as the ray makes one complete rotation around the plane (See Fig ). The words angle and rotation are synonymous with one another. An angle is measured from the ray s starting position along the positive x-axis (called its initial side) and ending at its terminal side. Angle Relationships and Degree Measurement Degree measurement is based on a circle, which is 360 degrees, or 360. If a ray is allowed to rotate one complete revolution counterclockwise around the xy-plane, starting and ending at the positive x-axis, we say that the ray has rotated 360 degrees. Its angle measurement would be written 360. Smaller (or larger) rotations can then be defined proportionally. For example, if the ray rotates half-way around the plane in the counterclockwise direction, from the positive x-axis to the negative x-axis, we say that the ray rotated 180 degrees (half of 360 degrees), and that its angle measurement is 180. Similarly, if the ray rotates from the positive x-axis to the positive y-axis in a counterclockwise direction, its rotation is 90 degrees and its angle measurement is 90.
2 Example 1: State the angle measurement in degrees for the following angles. 1. Initial side: positive x-axis, terminal side: negative y-axis, counter-clockwise rotation.. Initial side: positive y-axis, terminal side: positive x-axis, clockwise rotation. 3. Initial side: negative x-axis, terminal side: negative y-axis, counter-clockwise rotation. 4. Initial side: positive y-axis, terminal side: negative x-axis, clockwise rotation Solution: Some angles have special names: an angle of 180 is called a straight angle, and an angle of 90 is called a right angle. Angles between 90 and 180 are called obtuse and angles between 0 and 90 are called acute. When two angles sum to 180, they are called supplementary. When two angles sum to 90, they are called complementary. (See Fig 3). Fig 3 When two angles have the same initial and terminal sides, they are called coterminal. In fact, every angle has an infinite number of equivalent coterminal angles, since we may allow the ray to rotate many times about the plane, in either the positive or negative directions. For example, if a ray rotates counterclockwise from the positive x-axis to the positive y-axis, it forms an angle of 90. Equivalently, in the clockwise direction, this angle has a measure of 70. These two angles are coterminal; see Fig. 4: Fig. 4 If the ray was allowed to rotate more than once before ending at its terminal side, other coterminal angle measures would be 450, 810 and 1170 (one, two and three complete extra revolutions in the positive
3 direction), and 630, 990 and 1350 (one, two and three complete extra revolutions in the negative direction). In practice, we usually choose the smallest measure in most applications (See Fig 5). Fig 5 Example : Assume two rays meet, forming an angle of 7. State this angle s (a) supplement, (b) complement, (c) coterminal in the opposite direction, and (d) coterminal with one extra revolution in the positive direction. Solution: a) Its supplement is =153. b) Its complement is 90 7 =63. c) Its coterminal measure would be =33, but since it is traced in the opposite direction, it would have a negative value. Therefore, its coterminal measure in the opposite direction is 33. d) Its coterminal measure with one extra revolution in the positive direction is = 387. Example 3: A screw is turned 6 complete rotations, plus another half rotation, before it is set tight. What would be the angle through which this screw was turned? Solution: Six complete rotations would give an angle of 6 360=,160, and the extra half-rotation would be an extra 180. Thus, the screw was turned a total of,340. Example 4: Determine the following angles: 1. Two angles are supplementary, with the second angle being twice as large as the first angle.. Two angles are complementary, with the second angle being one less than three times the smaller angle. Solution: 1. Let be the smaller angle s measure. Therefore, the larger angle has a measure of. Since they are supplementary, their sum is 180. We solve for : +=180 3=180 =60 The smaller angle is 60 and the larger angle is 10.
4 . Letting be the smaller angle, the larger angle is 3 1, and their sum is 90. We solve for : +3 1=90 4 1=90 4=91 = 91 4 =.75 The smaller angle is.75 and the larger angle is Comment: the Greek letter (theta) is often used to represent an angle. Radian Measurement Suppose we overlay a circle with radius 1 on the xy-plane, its center at the origin. This is called the unit circle, and its circumference is =1=. A ray is drawn from the origin, intersecting the circle at point P (see Fig. 6). This ray is the terminal side of an angle, with the initial side being the positive x- axis, and this angle cuts off (subtends) a portion of the circle. The portion of the circle from (1,0) to point P (in the positive direction) is called an arc of the unit circle, and its length will be proportional to the circle s circumference (which is ) in the same way the angle s measure will be to 360 : Fig 6 length of arc radian measure π = angle in degrees 360 The length of the arc is called the radian measure of the subtending angle. A complete unit circle covers 360 and has a circumference of. Therefore, we can relate these two facts as follows: Dividing both sides by 360 gives us 360 degrees= radians 1 degree= radians = Or, we can divide by, which gives us degrees=1 radian. =
5 Therefore, we have two conversion formulas: 1. To convert from degrees to radians, multiply the degree measure by.. To convert from radians to degrees, multiply the radian measure by. Note: radian measure does not use the degree symbol, and often, the fraction is reduced but the is left alone. Example 5: Perform the following conversions. 1. Convert 100 into radians.. Convert radians into degrees. Solution: 1. We multiply 100 by and reduce: 100 =. Therefore, 100 is equivalent to radians.. We multiply by and reduce: = =3. Notice that =1. Therefore, radians is equivalent to 3. Some common angles can be quickly determined. These are very common and should be committed to memory: Half circle (straight angle): Quarter circle (right angle): Eighth circle: 180 = radians. 90 = radians. 45 = radians. The four quadrants can be defined in both degree and radian measurement. Let represent the value of the angle. Quadrant Degree Range Radian Range 1 0 < < 90 0<< 90 < < 180 << < < 70 << 4 70 < < 360 <<
6 Note the strict inequalities. Angles that are exactly 0,90,180,70 or 360 (in radians: 0,,, or ) are called cardinal angles. They are not part of any quadrant. The Trigonometric Functions Let us look again at a unit circle (radius = 1) placed on an xy-axis system with its center at the origin. A ray is drawn, intersecting the circle at point P as shown in Fig. 7. Let represent the angle from the positive x-axis to the ray. Three definitions can now be made, each corresponding to a physical characteristic of the ray and its intersection of the unit circle at point P: DEFINITION: The cosine of, written cos, is the x-coordinate of P. The sine of, written sin, is the y-coordinate of P. The tangent of, written tan, is the slope of the ray. Fig. 7 With these definitions, some immediate results can be determined for the cardinal angles based on the unit circle: For 0 (0 radians), this is the point (1,0). Therefore, cos0=1,sin0=0 and the slope of the ray is tan0=0. For 90 ( radians), this is the point (0,1). Therefore, cos90=0, sin90=0 and since the ray is vertical, the slope is not defined (that is, tan 90 is not defined). For 180 ( radians), this is the point (-1,0). Therefore, cos180= 1, sin180=0 and the slope of the ray is tan180=0. For 70 (3 radians), this is the point (0,-1). Therefore, cos70=0,sin70= 1 and the slope of the ray is undefined since it is vertical.
7 For most other angles, a calculator is necessary to calculate the cosine, sine and tangent of an angle. Example 6: A ray with an angle of 50 is drawn, with initial side being the positive x-axis. State the point at which this ray intersects the circle, and the slope of this ray. Solution: Using a calculator set to Degree mode, the x-coordinate is cos50=0.643 (rounded), and the y-coordinate is sin50=0.766 (rounded). The slope of the ray is tan50=1.19 (rounded). We can also observe some geometrical relationships between sine, cosine and tangent. First, we can view the portion of the ray from the origin to the circle as a hypotenuse (of length 1) of a right triangle. The legs have length cos (called the adjacent leg) and sin (called the opposite leg) (See Fig. 8). Fig 8 Therefore, we can use the Pythagorean formula to form a relationship between cosine and sine: This is usually written as adjacent leg +opposite leg =hypotenuse cos +sin =1 cos +sin =1.
8 This identity is called the Pythagorean Identity. Notice that we write cos as a short-hand for cos. Another very common relationship between sine, cosine and tangent uses the slope formula. Since the ray passes through the origin (0,0) and point P, which is cos,sin, we can calculate the slope of the ray, which is defined to be tan: Therefore, we have Trigonometric Values of Special Angles Slope of ray=tan= sin 0 cos 0 tan= sin cos Exact values for the sine, cosine and tangent of some special angles can be determined using geometry. We look at the case of 45 radians first: Suppose the ray is drawn with an angle of 45 on the unit circle as seen in Figure 9 below. Let P be its intersection with the circle. Fig. 9 The hypotenuse has length 1, and its x and y coordinates will be the same. Using the Pythagorean formula, we have + =1 Remember, x = y. =1 = = =. Rationalizing the denominator. The x-coordinate of P is, and the y-coordinate of P is also. Therefore, cos45= Since the slope is 1, we observe that tan45=1. and sin45=.
9 Using radians, we have cos =, sin = and tan =1. Now suppose we draw the ray at an angle of 60 radians, intersecting the unit circle at point P. Recall from geometry that an equilateral triangle (all sides equal in length) has internal angles of 60. Therefore, we draw a line from P to the point (1,0) and form an equilateral triangle with sides of length 1. Now divide this triangle in half by sketching a vertical line from P to the point,0 (See Fig 10). Fig. 10 We now have a right triangle with hypotenuse of length 1 and an adjacent leg of length. Using the Pythagorean formula, the opposite leg has length. Point P s coordinate is,. and we have The slope of this ray is cos60= tan60= and sin60=. = =3. / In radians, we have cos =, sin = and tan =3. A similar construction shows that cos30=, sin30= cos =, sin = and tan =. This is left as homework exercise XX. These values are summarized in the following table: Angle Angle cos sin tan (Degrees) (Radians) / / undefined Note that these values are for the first quadrant. and tan30=. In radians, we have
10 In Quadrant, the cosine will be negative (since the x-coordinate of any point in Quadrant is negative). The sine will stay positive, and since rays in Quadrant have negative slope, the tangent will be negative. In Quadrant 3, both the cosine and sine values are negative, but rays in this quadrant have positive slope, so the tangent is positive. In Quadrant 4, the cosine is positive, the sine is negative and the tangent is negative. This is summarized in the following figure: Remember, All Students Take Calculus. In Quadrant 1, all trigonometric functions are positive. In Quadrant, the sine function is positive. In Quadrant 3, the tangent function is positive, and in Quadrant 4, the cosine function is positive. We use symmetry to determine the actual values for cosine, sine and tangent. Example: Suppose =10. Determine cos,sin and tan. Solution: Since 10 is in Quadrant, the cosine will be negative, the sine positive and the tangent negative. The ray that represents 10 is drawn, and we notice that it is symmetric across the y-axis with the ray for 60. We know from the above table that cos60=, sin60= and tan60=3. We attach a negative sign to the cosine and tangent values, and leave the sine value alone. Therefore, cos10=, sin10= and tan10= 3. In the above example, the 60 was the reference angle used to determine the trigonometric values for 10. The reference angle lies in Quadrant 1, and to determine the correct reference angle for an angle in Quadrants, 3 or 4, refer to the table below: Quadrant Reference Angle Formula Symmetry 180 (degrees) Across y-axis (radians) (degrees) Across origin (radians) (degrees) (radians) Across x-axis
11 Graphs of the Trigonometric Functions If we treat as a variable representing the angle of a ray as it rotates counterclockwise in the xy-plane, we may then treat the cosine, sine and tangents as functions of : =cos =sin h=tan Note: when graphing any trigonometric function, the angle (input) is always in radians. Graph of the function =cos. We can create a table of values for this function, as makes one complete positive rotation about the xy-plane. cos0 =1 cos6 =3 cos4 = cos3 =1 cos =0 cos3 = 1 cos34 = cos56 = 3 cos = 1 cos76 = 3 cos54 = cos43 = 1 cos3 =0 cos53 =1 cos74 = cos116 =3 cos=1 Plotting these points and connecting them with a smooth curve generates a portion of the cosine graph: The complete graph of =cos is defined for <<. Notice that the wave repeats itself every units. The range of =cos is 1 cos 1. Graph of the function =sin In a similar manner, the graph of the function =sin is generated
12 It also repeats itself every units, and it too has a range of 1 sin 1. The graph of the function h=tan The tangent function is undefined when =±,±,±,. These angles will force the ray to be vertical, in which case the slope of the ray is undefined. On the graph, there will be vertical asymptotes at these values. Other values can be generated using methods we have discussed previously. The complete graph of the tangent function is: The range of the tangent function is <tan<. The portion of the tangent function between << is called the main branch of the tangent function. These branches repeat forever, with a period of units.
13 Periodicity Since the ray can rotate infinitely often, the function (output) values will repeat in a consistent pattern. In the case of the cosine and sine functions, these values will repeat every time the ray has completed a full rotation, that is, it has swept out an angle of radians. Therefore, we have This leads us to a definition of periodicity. DEFINITION cos+=cos sin+=sin A function is periodic if there exists a value p such that +=, for all x in the domain of f. The smallest such value of p is called the period of the function. Therefore, the cosine and sine function each have period (If the angle is given in degrees, the period is 360 ). The tangent function has a period of units. That is, tan+=tan. These facts can be used to reduce large angles into smaller equivalents, as the next example shows: Example: Determine the following measurements: 1. sin. cos 3. tan Solution: 1. The angle radians can be reduced by recognizing that it is +. Therefore, sin =sin +=sin. We know that sin =. Therefore, sin =.. The angle is the same as +4. The 4 represents two periods, or two complete rotations of the angle through the xy-plane. We note that what remains, the, is in Quadrant 3, and that the reference angle is. We know that cos =, and since the cosine is negative in Quadrant 3, we conclude that cos =. 3. The angle is the same as +. Therefore, tan =tan +=tan.
14 Since tan =, we conclude that tan =. Right-Triangle Trigonometry and the Reciprocal Functions The sine, cosine and tangent functions can be defined on a right triangle. Let angle be placed as shown in Fig. 11. Fig. 11 Remember, the leg closest to this angle is called the adjacent leg, and the other leg is called the opposite leg. Therefore, the sine, cosine and tangent functions can be defined in terms of the lengths of the adjacent leg, the opposite leg and the hypotenuse: sin=, cos=,tan=. Using O for opposite, A for adjacent and H for hypotenuse, the mnemonic SOHCAHTOA is a useful way to commit these relationships to memory. Often, the trigonometric values can be determined even if the angle is unknown, as the following example shows. Example: A right triangle as an adjacent leg of length 7 and a hypotenuse of length 10. Determine exact values for sin, cos and tan. Solution: We use the Pythagorean Formula to determine the length of the opposite leg (denoted y): Therefore, sin= = 7 + = =100 =51 =51, cos= = and tan= =. The reciprocal functions are defined as follows:
15 DEFINITION The secant of, written sec, is the reciprocal of cos. That is, sec= =. The cosecant of, written csc, is the reciprocal of sin. That is, csc= =. The cotangent of, written cot, is the reciprocal of tan. That is, cot=. = = The physical interpretations (as lengths) of all six trigonometric functions can be seen on one diagram, as shown below. Example: Suppose sin= and that is in Quadrant 1. Determine the remaining five trigonometric function values of in exact (non-decimal) form. Solution: In Quadrant 1, all six trigonometric functions are positive. Since sin=, we know the opposite leg has length 4 and hypotenuse has length 7. The Pythagorean Formula is used to determine the length of the remaining (adjacent) leg: +4 =7 +16=49 =33 =33 Since cos=, we conclude that cos=. Also, since tan=, we conclude that tan= =, where the denominator was rationalized in the last step. The remaining trigonometric function values are determined by taking appropriate reciprocals: csc= sec= = = ; = = Rationalize the denominator
16 cot= = =. Rationalize the denominator Had the angle been in a different quadrant, negative signs would have been assigned according to the ASTC mnemonic. For example, if was in quadrant in the above example, the cosine and tangent results would have been negative, and their reciprocal (the secant and cotangent) results would have been negative as well. The cosecant result would have remained positive. For most practical measurement purposes, the cosine, sine and tangent functions are almost always sufficient. However, in calculus, the reciprocal functions (in particular, the secant function) have many uses. Common Identities There are many useful identities that we use in trigonometry and in calculus. The most common involving the cosine and sine functions are stated below. The Sum and Difference Identities sin±=sincos±cossin cos±=coscos sinsin Note the reversal of the signs in the cosine sum-difference formula. Example: Use an appropriate sum-difference formula to determine the exact value of sin75. Solution: We see that 75 = Therefore, sin75 =sin =sin45 cos30 +cos45 sin30 = + Referring to the table of 1st Quadrant values = These sum-difference identities can be used to create the double-angle identities: The Double Angle Identities sin=sincos cos=cos sin The proofs of both are left as exercises XX and YY in the homework. Example: If cos= and is in Quadrant 1, determine exact values for sin and cos. Solution: We ll need the exact value for sin, which we can get from the Pythagorean Identity it is sin= (You should verify this).
17 Therefore, sin=sincos = = Also, we have cos=cos sin = = = Why is cos negative? In this case, since was in the 1 st Quadrant, doubling its size evidently placed the angle into Quadrant, where the cosine is negative and the sine remains positive. The Pythagorean Identity is cos +sin =1. Re-arranging or dividing through by terms produces many corollary forms of the Pythagorean Identity. The Pythagorean Identities and Corollaries cos +sin =1 cos =1 sin (subtraction of sin ) sin =1 cos (subtraction of cos ) 1+tan =sec (divide through by cos ) tan =sec 1 (subtraction of 1) cot +1=csc (divide through by sin ) cot =csc 1 (subtraction of 1) Example: The double angle identity for the cosine function is Show that this formula can also be written as cos=cos sin. 1 cos=1 sin cos=cos 1 Solution: In the first case (1), we substitute cos with 1 sin and simplfy:
18 cos=cos sin =1 sin sin =1 sin In case (), we substitute sin with 1 cos and simplify: Homework Set cos=cos sin =cos 1 cos =cos 1 State the angle in degrees. 1. Initial side: positive x-axis, terminal side: negative x-axis, clockwise rotation.. Initial side: negative y-axis, terminal side: positive x-axis, counterclockwise rotation. 3. Initial side: positive y-axis, terminal side: positive y-axis, counterclockwise rotation. 4. Initial side: negative x-axis, terminal side: negative y-axis, clockwise rotation. For the given angles, state (a) whether it is acute, obtuse or neither, (b) its supplement, (c) its complement, if it is defined, and (d) its quadrant, if it is defined For the given angles, state its equivalent coterminal angle as described , negative rotation , negative rotation , one extra positive rotation , one extra positive rotation , one extra negative rotation , one extra negative rotation , two extra positive rotations , three extra negative rotations. 1. A basketball makes 5 complete rotations between the time it is let go by the shooter and before it enters the basket. How many degrees did the basketball turn while in the air?. A figure skater makes 8 complete rotations plus an extra half-rotation during part of her routine. How many degrees did she rotate?
19 3. Two angles are complementary, and one angle is 4 times as large as the smaller angle. State the two angles. 4. Two angles are complementary, and the smaller angle is 1/5 as big as the larger angle. State the two angles. 5. Two angles are supplementary, and the larger angle is 3.4 times as big as the smaller angle. State the two angles. 6. Two angles are supplementary, and the larger angle is 5 degrees less than twice the size of the smaller angle. State the two angles. Convert the following angles from degrees into radian measure Convert the following angles from radian measure into degrees State the complement and supplement to in radian measure.
20 5. State the complement and supplement to in radian measure. 53. How many degrees is 1 radian? 54. An angle has a radian measure of 4 radians. Which quadrant is this angle located in? 55. An angle sweeps out 1/10 th of a circle in the positive direction. What is this angle s radian measure? 56. An angle sweeps out /5 th of a circle in the positive direction. What is this angle s radian measure? In the following problems, a ray is drawn at the given angle, intersecting the unit circle at point P. State P s coordinates and the ray s slope. Use a calculator and state your answers to three decimal places Use the Pythagorean Identity, quadrants and symmetry to answer the following questions. State your answers to three decimal places. 65. If sin=0.15 and is in Quadrant 1, find cos and tan. 66. If sin=0.70 and is in Quadrant 1, find cos and tan. 67. If cos= and is in Quadrant, find sin and tan 68. If cos= and is in Quadrant, find sin and tan 69. If sin= and is in Quadrant 3, find cos and tan 70. If cos= and is in Quadrant 3, find sin and tan 71. If cos=0.51 and is in Quadrant 4, find sin and tan 7. If sin= and is in Quadrant 4, find cos and tan Use reference angles and symmetry to answer the following questions in exact form. 73. sin 74. sin 75. cos 76. cos 77. sin
21 78. cos 79. cos 80. sin Determine the remaining five trigonometric function values in exact form, based on the given information. 81. sin=, is in Quadrant sin=, is in Quadrant cos=, is in Quadrant. 84. sin=, is in Quadrant. 85. tan=, is in Quadrant sin=, is in Quadrant sin=, is in Quadrant tan= 5, is in Quadrant 4. Use your calculator to determine these trigonometric values to three decimal places. Note: most calculators do not have specific keys for the cotangent, secant and cosecant. 89. cot5 90. cot csc11 9. csc sec sec If sin= and is in Quadrant 1, determine sin and cos. 96. If cos= and is in Quadrant, determine sin and cos. 97. Use a sum-difference formula to prove sin=sincos. 98. Use a sum-difference formula to prove that cos=cos sin 99. The Shift Identities are cos =sin sin+ =cos Use the sum-difference formulas to prove these two identities.
22 100. Use the sum-difference identities to show that sin30=, cos30= tan30=. Hint: 30 = and 101. What is the period of the graph of =sin? 10. What is the period of the graph of =sin? 103. The population of a mining town is modeled by the function = cos 1, Where t is months (January = 1, so forth). Your calculator should be in Radian mode. a) Calculate the town s population in March, July and November. b) In what month is the population the lowest? The highest?
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
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
Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is
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 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
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
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
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
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
Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions A unit circle is a circle with radius = 1 whose center is at the origin. Since we know that the formula for the circumference
About Trigonometry TABLE OF CONTENTS About Trigonometry... 1 What is TRIGONOMETRY?... 1 Triangles... 1 Background... 1 Trigonometry with Triangles... 1 Circles... 2 Trigonometry with Circles... 2 Rules/Conversion...
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
5.2 Trigonometric Functions Of Real Numbers Copyright Cengage Learning. All rights reserved. Objectives The Trigonometric Functions Values of the Trigonometric Functions Fundamental Identities 2 Trigonometric
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
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
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.
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
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
Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the
Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving real-world problems and in the development of mathematics. Whatever their use, it is often of value
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
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:
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,
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
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,
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
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
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
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
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
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
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,
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
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
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)
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
(cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each
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
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:
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
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
CHAPTER 1 The Six Trigonometric Functions Copyright Cengage Learning. All rights reserved. SECTION 1.1 Angles, Degrees, and Special Triangles Copyright Cengage Learning. All rights reserved. Learning Objectives
Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use
Lecture Two Trigonometric Section.1 Degrees, Radians, Angles and Triangles Basic Terminology Two distinct points determine line AB. Line segment AB: portion of the line between A and B. Ray AB: portion
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
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
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:
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
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
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
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Pre-Calculus Unit : Trigonometric Functions These materials are for nonprofit educational purposes only. Any other use may constitute
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
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
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
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
Exponents An integer is multiplied by itself one or more times. The integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself. Example:
Learning Assistance Center Southern Maine Community College ANGLE CALCULATIONS An angle is formed by rotating a given ray about its endpoint to some terminal position. The original ray is the initial side
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
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
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56
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,
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,
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
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
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
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
Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an
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
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
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
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
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
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
Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope
P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary
Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane
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
1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra
Moore Catholic High School Math Department COLLEGE PREP AND MATH CONCEPTS The following is a list of terms and properties which are necessary for success in Math Concepts and College Prep math. You will