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THAT’S TRIGONOMETRY For information: Fred W. Duckworth, Jr. c/o Jewels Educational Services 1560 East Vernon Avenue Los Angeles, CA 90011-3839 E-mail: [email protected] Website: www.trinitytutors.com Copyright © 2008 by Fred Duckworth. All rights reserved. This publication is copyrighted and may only be copied, distributed or displayed for personal use on an individual, one-time basis. Transmitting this work in any form by any means, electronic, mechanical or otherwise, without written permission from the publisher is expressly prohibited. All copyright notifications must be included and you may not alter them in any way. Moreover, you may not modify, transform, or build upon this work, nor use this work for commercial purposes. 2 Lesson 1 Why Study Trigonometry? __________________________________________________________________ Trigonometry – which was originally all about the relationships that exist between the sides of a right triangle with respect to a given angle – has become an indispensable tool among architects, surveyors, astronomers, navigators, and others. This is especially true when it comes to spherical trigonometry, which is concerned specifically with the study of triangles on the surface of a sphere such as the Earth. We can use trigonometry to determine distances and heights that cannot easily be measured directly, such as the height of an unscalable mountain, the altitude of a plane flying overhead, or the width of a river too deep to ford. Astronomers also use it to calculate distances to objects in space, and there exist many other much more complex applications as well. Most importantly for us, we will need to learn trigonometry in order to understand calculus in the future. (Calculus is used to calculate the area or volume of shapes or figures that have curves which change at constantly varying degrees, or to calculate a moving object’s varying rate of speed at a given point along its path based on the moment in time it was there.) __________________________________________________________________ A PRACTICAL APPROACH Trigonometry uses the techniques that you previously learned from the study of algebra and geometry. We will introduce and define the trigonometric functions you will need to learn geometrically rather than in terms of algebraic equations since this is the way they were originally derived, and we believe virtually all math topics are most easily understood when presented exactly as they were discovered by their originators. We will go on to help you develop facility with these functions as well as the ability to prove basic identities regarding them. This is especially important if you intend to study calculus, more advanced mathematics, physics and other sciences, or engineering. One of the things that the state of California wants you to know are the definition of sine and cosine as y- and x- coordinates of points on a unit circle, which is probably about the best place to start, so here we go. 3 Lesson 1 The Definitions We Are Looking for a “Sine” Trigonometry is a specialized area of geometry that began by quantifying the relationships—or ratios—between each side of a right triangle. The term is derived from the Greek word trigonon, meaning triangle, and from the Greek word metron, meaning measurement. The six relationships just mentioned are best conceptualized using something called a unit circle, which is a circle that has a radius of one unit and has its center at the origin of an x-y rectangular coordinate system or plane. (See the illustration below.) Inscribed inside of our unit circle we have a unit triangle, which is a right triangle formed by the x-axis, along with a diagonal line that meets the x-axis at the origin to form central angle θ (theta), and finally, the top half of a chord that is bisected by the x-axis. (The bottom half of the chord not pictured. You may recall that a chord is a line segment that joins two points on a curve.) 4 We call the top half of the chord that is bisected by the x-axis (the bottom half of the chord is not pictured) sine, a term that is derived from a word meaning “half chord.” sine For reasons we will spell out later, we call the other side of our right triangle the cosine. 5 Going back to our unit-circle, let tangent be a line segment parallel to sine and tangent to the side of the circle. let tangent be a line segment parallel to cosine and tangent to the top of the circle. 6 The word secant is from the Latin root secare, meaning “to cut.” The name is fitting for a segment that cuts through the circle. We will color secant maroon for easy identification. The cosecant is the line segment that continues to cut through the circle until it unites with the cotangent. 7 Lesson 2 The Trigonometric Functions __________________________________________________________________ Note that the x-axis, the radius, and sine all form a triangle inside of our unit circle—one in which the hypotenuse is also the radius. Note also that the side adjacent to angle θ lies along the x-axis, and that the (x, y) coordinates at the outer end of the hypotenuse identify the lengths of the sides adjacent to and opposite of angle θ respectively. As already mentioned, such a triangle is called a unit-triangle. Our goal now is to “define” the six relationships, or ratios, mentioned earlier, beginning with sine. To create our first “definition,” we need to find another part of our unittriangle that we can relate to sine in such a way as to end up with a number that is equal to the length of line segment we have given the name: sine. We can accomplish this by relating the length of sine to any other side of the triangle that equals one. So, dividing the length of sine by the hypotenuse will give the length of sine right back to us. But, be careful! Remember that trigonometry was originally all about the relationships that exist between the sides of a right triangle with respect to a given angle. So, although we initially labeled sine as the half-chord directly across from angle θ, in actuality, sine is more accurately described as . . . 8 The ratio between the half-chord directly across from angle θ and the hypotenuse (radius). Note that the length of “sine” is equal to y, so what we have is … y r So, sine is not really the opposite side, but rather, the relationship between side opposite and the hypotenuse. And since the half-chord (the line segment we've been calling sine) is opposite angle θ, in the final analysis what we actually have is . . . sin sin y r THE TRIGONOMETRIC FUNCTIONS sin y r csc r y cos x r sec r x tan y x cot x y If you want to memorize all of the algebraic functions as virtually every other trigonometry course requires you to do, here are links to a couple of sites that will help you do so. However, if you want to make memorization unnecessary by being able to literally “see” what each one of the functions “means,” set aside some time to study the illustrations on 9 the previous pages, spending the time it takes to be able to visualize the relationships listed below in your “mind’s eye” so that it all comes together making perfectly logical sense and isn’t just a collection of abstract facts that have absolutely no meaning to you. REVIEW The tangent is like the side opposite divided by the side adjacent, which would be equivalent to sine divided by cosine. The cotangent is like the side adjacent divided by the side opposite, which is equivalent to 1 divided by the tangent. That means that cotangent is the reciprocal of tangent. (It is also like side adjacent divided by side opposite.) This secant is like the hypotenuse divided by the side adjacent, which is equivalent to 1 divided by cosine. That means that the secant is the reciprocal of cosine. And finally, the cosecant is like the hypotenuse divided by side opposite, which is the same as 1 divided by sine. This means that the cosecant is the reciprocal of sine. As you can see, sine corresponds to the side opposite theta. If we divide the side opposite by the hypotenuse (which equals one) what we get back is identical to what we started with: the side opposite. So, now we have our first ratio. sin opposite hypotenuse (We abbreviate sine with sin.) 10 We can do the same thing in the case of cosine, which corresponds with the side that is adjacent to angle theta. cos adjacent hypotenuse (We abbreviate cosine with cos.) Now that we have related both the side opposite theta and the side adjacent to theta to our hypotenuse, it is time to relate them to each other. The only problem is, to form ratios in which the denominators will be the numeral 1 we are going to have to use a couple of similar triangles, the first of which is pictured below (in yellow). 11 As you can see, the side adjacent to theta is equal to one (the radius), so by dividing the side opposite (which corresponds to tangent) by side adjacent (which equals 1) we get back the same thing we started with, which is side opposite (or tangent). So we have tan opposite adjacent To form a ratio that has cotangent as the numerator and 1 as the denominator we are going to have to use a second similar triangle (once again filled in with yellow). If we were to manipulate the triangle by making two turns counterclockwise, we would find that the side that is equal to 1 (the radius) would correspond to side opposite, and cotangent would correspond to side adjacent, so that we have… cot adjacent . opposite 12 y r=1 x So then, when it comes to defining our ratios, both sine and tangent correspond to side opposite, while cosine and cotangent correspond to side adjacent. Now all that remains is relating the hypotenuse (which corresponds to secant and cosecant) to side opposite and side adjacent. To divide the hypotenuse of our first similar triangle (which corresponds to secant) by 1 we need to form a ratio with side adjacent, so we have… sec hypotenuse . adjacent 13 And to divide the hypotenuse of our second similar triangle (which corresponds to cosecant) by 1 we need to form a ratio with side opposite, so we have… csc hypotenuse . opposite 14 The resulting six ratios constitute… THE TRIGONOMETRIC FUNCTIONS OF RIGHT ANGLES sin opposite hypotenuse csc hypotenuse opposite cos adjacent hypotenuse sec hypotenuse adjacent tan opposite adjacent cot adjacent opposite However, since the length of side adjacent is equal to x, and the length of side opposite is equal to y, with the length of the hypotenuse equal to r (radius), we could write the trigonometric functions of right angles using these three variables instead. However, if we do, we no longer have the trigonometric functions of right angles. We now have… 15 Then again, since side adjacent corresponds to cosine, and side opposite corresponds to sine, while the hypotenuse corresponds to Special Angles 45° You need to know the function values of certain special angles and you really need to memorize them because you’ll use them so often that deriving them or looking them up every time would really slow you down. Let's begin with Functions of 45°. Look at the 45°-45°-90° unit-triangle illustrated below. Figure 1.9 Y r=1 θ x 16 Since the complementary angles are equal, the sides opposite each (sine and cosine) will also be equal. By the Pythagorean theorem, sine² + cosine² = (hypotenuse)² Let a = the length of sine. Since sine = cosine, a also = the length of cosine And if hypotenuse = 1, then (hypotenuse)² = 1² = 1 So again, by the Pythagorean theorem we have: a² + a² = 1 2a² = 1 a² = ½ a = √ ½ = 1/√ 2 = (√2 ) / 2 Since a = sin 45° = cos 45°, sin 45° = cos 45° = (√ 2 ) / 2 Also, b = cos 45° and b = a; therefore cos 45° = (√2)/2 And tan 45° = side opp / sid adj = cos 45° / sin 45° = 1 17 Special Angles 30°60° 18 Lesson 8 Measuring Angles The state of California expects you to understand the notion of an angle and how to measure it, in both degrees and radians. You are also expected to be able to convert between degrees and radians, so let’s start by developing a clear understanding of degrees. As you almost assuredly know, a day is measured in units called hours, each of which is divided into 60 minutes. Similarly, angles are measured in units called degrees, each of which is also divided into 60 minutes. And just as each minute of time is divided into 60 seconds, each minute of a degree is also divided into 60 seconds. So, just as a second of time is 601 of a minute…and a minute of time is 601 of an hour, so it is when measuring degrees of an angle. Traditionally, portions of a degree have been measured in minutes and seconds. One minute is 601 of a degree and is written like this: 1'. Moreover, one second is 1 60 of a minute and is written like this: 1". Consequently, the measure 12° 42' 38" represents 12 degrees, 42 minutes, 38 seconds. CALCULATING WITH MINUTES AND SECONDS To add angle measures that not only involve degrees, but minutes and seconds as well, you have to add the degrees, minutes and seconds separately. CONVERTING DECIMAL DEGREES 19 Also, with the increasing use of calculators, it is now common to measure angles in decimal degrees. To convert minutes and seconds into decimal degrees, divide the minutes by 60, and the seconds by 3600. To convert decimal degrees into minutes and seconds, you obtain the minutes by multiplying the decimal portion of the measure times 60. (The whole part of the result will be the number of minutes.) To obtain the number of seconds, multiply the decimal part of the result times 60 as well. EXAMPLE: Convert 34.817° to degrees, minutes, seconds. Convert 34.817° to degrees, minutes, seconds. 34.817° = 34° + .817° = 34° + (.817)(60') = 34° + 49.02' = 34° + 49' + .02' = 34° + 49' + (.02)(60") = 34° + 49' + 1" (rounded) = 34° 49' + 1" terminal side initial side 20 The Pythagorean Identities THE PYTHAGOREAN THEOREM You know that according to the Pythagorean Theorem a2 + b2 = c2. Well, returning to our unit triangle, we can see that x2 + y2 = r2 Figure 1.1 Since x2 corresponds to cosine, y2 corresponds to sine, and r2 corresponds to 1, we have cos2 θ + sin2 θ = 1. Because the Pythagorean theorem is used to get x2 + y2 = r2 this equation is referred to as a Pythagorean Identity and is written… sin2 θ + cos2 θ = 1 If we divide both sides by sin2 we get, sin 2 cos 2 1 2 2 sin sin sin 2 sin 2 cos 2 1 Of course, 2 1 , cot 2 , and csc 2 , which means that… 2 2 sin sin sin 1 + cot2 θ = csc2 θ 21 And finally, if we divide both sides by cos2 we get, Of course, sin 2 cos 2 1 2 2 cos cos cos 2 sin 2 cos 2 1 2 tan , 1 , and sec 2 , which means that… 2 2 2 cos cos cos tan2 θ + 1 = sec2 θ Thus we have the three Pythagorean Identities. The state of California expects you to be able to use the Pythagorean theorem to prove the fact that the identity, cos2 (x) + sin2 (x) = 1 is equivalent, as well as prove the Pythagorean Theorem as a consequence of this identity, so take the time right now to make sure that you understand how to perform the necessary calculations on your own, without any assistance from your notes, the textbook, or any other resource. You are also expected to prove other trigonometric identities and simplify them by using the identity cos2 (x) + sin2 (x) = 1. For example, you may be asked to use this identity to prove that sec2 (x) = tan2 (x) + 1, so make sure you understand how to perform the necessary calculations to generate the other two Pythagorean Identities as well. RECIPROCAL INDENTITIES (PAGE 31) 22 Lesson 8 Graphing Functions page 112 The 4.0 Students graph functions of the form f(t) = A sin ( Bt + C ) or f(t) = A cos ( Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift. 23 Lesson 8 Graphing Functions page 112 The 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. 6.0 Students know the definitions of the secant and cosecant functions and can graph them. 7.0 MEMORIZE The tangent of the angle that a line makes with the x-axis is equal to the slope of the line. (Find out what this means.) 24 Lesson 8 Inverse Trigonometric Functions Page 207 INVERSE TRIGONOMETRIC FUNCTIONS page 207 Function Domain Range y = Sin –1 x y = Cos –1 x y = Tan –1 x y = Cot –1 x y = Sec –1 x y = Csc –1 x –1 ≤ x ≤ 1 –1 ≤ x ≤ 1 Any real number Any real number x ≤ –1 or x ≥ 1 x ≤ –1 or x ≥ 1 /2 y /2 0 y – /2 < y < /2 0<y< 0 y < /2 or y < 3 /2 * 0 < y /2 or < y 3 /2 * 8.0 Students know the definitions of the inverse trigonometric functions and can graph the functions. 25 9.0 Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. Trigonometric Formulas Addition Formulas for Sines and Cosines Page 172 10.0 The state of California expects you to be able to demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/ or simplify other trigonometric identities. SUMS AND DIFFERENCES OF SINE AND TANGENT sin( A B ) sin A cos B cos A sin B sin( A B ) sin A cos B cos A sin B tan( A B) tan A tan B 1 tan A tan B tan( A B ) tan A tan B 1 tan A tan B 26 Trigonometric Identities Half-Angle Identities Page 183 11.0 Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and use those formulas to prove and/ or simplify other trigonometric identities. cos 1 cos A A 2 2 sin 1 cos A A 2 2 tan A 1 cos A 2 1 cos A tan A sin A 2 1 cos A tan A 1 cos A 2 sin A 27 Trigonometric Identities Double-Angle Identities Page 177 11.0 The state of California expects you to be able to demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and use those formulas to prove and/ or simplify other trigonometric identities. cos 2 A cos 2 A sin 2 A cos 2 A 1 2 sin 2 A cos 2 A 2 cos 2 A 1 sin 2 A 2 sin A cos A 2 tan A tan 2 A 1 tan 2 A 28 Lesson 8 Right Triangles Page 17 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. 29 Triangles and Vectors The Trigonometric Laws LAW OF SINES If any triangle ABC, with side a, b, and c, a b , sin A sin B a c , sin A sin C and b c . sin B sin C See Chapter whatever for practice problems using the Law of Sine. THE LAW OF COSINES 13.0 Students know the law of sines and the law of cosines and apply those laws to solve problems. PAGE 231 PAGE 243 DETERMINING AREA 14.0 Students determine the area of a triangle, given one angle and the two adjacent sides. PAGE 234, 245 30 Polar Equations Polar Coordinates Pages 283, 284 15.0 Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa. 16.0 Students represent equations given in rectangular coordinates in terms of polar coordinates. 31 Lesson 8 Complex Numbers There is no real number solution to the equation x2 + 1 = 0. For such problems, we must turn to a set of numbers that has real numbers as a subset, that is, the set of complex numbers. The state of California and not only expects you to be familiar with complex numbers, but to also be able to represent a complex number in polar form and know how to multiply complex numbers in their polar form. OPERATIONS ON COMPLEX NUMBERS Complex numbers have the form a + bi, where a and b are real numbers and i is the new number defined by i 1 or 32 i2 = –1 Each real number is a complex number, since a real number a may be thought of as the complex number a + 0i. A complex number of the form 0 + bi, where b is nonzero, is called an imaginary number. Both the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. However, they share no members in common. Moreover, complex numbers that are neither imaginary numbers nor real numbers can only be written in standard form—with two terms and an operation sign, either the plus sign (+) or the minus sign (–) . Lesson 8 DeMoivre’s Theorem The 18.0 Students know DeMoivre's theorem and can give n th roots of a complex number given in polar form. If r(cos θ + i sin θ) is a complex number and if n is any real number, then [r(cos θ + i sin θ)]n = r n(cos nθ + i sin nθ). 33 Lesson 8 Solving Problems Using Trigonometry Page 233 Jean Johnson wishes to measure the distance across the Big Muddy River. She finds that C = 112° 53', A = 31° 06', and b = 347.6 ft. Find the required distance. 34 We can use the law of sines to find a, but first, we must find angle B. 19.0 Students are adept at using trigonometry in a variety of applications and word problems. 35