Trigonometry 9.1 Radian and Degree Measure Angle Measures I am aware of three ways to measure angles: degrees, radians, and gradians. In all cases, an angle in standard position has its vertex at the origin, one ray (the initial side) along the positive x-axis, and the other ray (the terminal side) pointing in some direction. Degrees divide a circle into 60 parts, each of which is called a degree. These are very familiar to most students. The symbol to denote a degree is a small circle in the superscript position: 1º. Gradians divide a circle into 400 parts, so that a right angle is 100 gradians. I have only seen this used in military applications. Radians don't divide the circle into parts; they measure the size of the central angle in a sector of a unit circle with a certain arc length (whew!). Thus, radians really measure the length of an arc. The central angle measures one radian r r Arc Length of r Figure 1 - Radian Measure In mathematics, the radian is the preferred (and in many cases, only) option. In applied situations, degrees are preferred although certain applications require that radians be used; which forces conversions at the beginning and end of the problem! You should be able to convert between degrees and radians. Since radians deal with arc length, then a full circle angle (60º) is equal to a number of radians that is the same as the circumference (πr). The angle part of that is just π just remember that 180º = π (radians), and you'll always be able to set up a proportion to convert between degrees and radians. Arc Length The length of the arc subtended by an angle of measure θ in a circle of radius r is AL = rθ (provided the angle is measured in radians). If it isn't, then you need to set up a proportion: AL θ π r = (where the angle is now in degrees). Of course, this is the same thing as just 60 converting to radians, and then using the easier formula Other Topics A long time ago, you worried about angles that were acute, right, and obtuse we'll not be terribly concerned with these definitions. Similarly, we'll not really deal with supplementary and complementary angles. CAIC NOTES 09, PAGE 1 OF 11
[1.] Convert 18º to radians. π π 18 =. 180 10 [.] Convert 4 π to degrees. 4π 180 = 4 6 = 144. π [.] Find the length of the arc subtended by an angle of 1º in a circle of radius 7. π π π 7π 1 = ; AL = rθ = 7 =. 180 1 1 1 9. The Unit Circle Definition The unit circle is a circle of radius one, centered at the origin. Many students have dark nightmares about the unit circle, mainly because they don't understand what it's for: a memorization tool. The special right triangles that you (should have) learned in Geometry do exactly the same thing! Forget all this "wrapping" stuff; it doesn't help. Special Right Triangles This knowledge is essential. Through the unit circle, or special right triangles, knowing the relationships between angles and lengths/coordinates is a must. 0-60-90 0º x x x 60º CAIC NOTES 09, PAGE OF 11
4-4-90 x 4º x x The Connection As it turns out, every angle intersects the unit circle at a point, which creates a right triangle with hypotenuse of length 1. That is useful once you know how to deal with right triangles and points in space so keep reading, and it will all become clear eventually. Of course, that's why I'm saving this section for later, as we actually take notes. 9. Right Triangle Trigonometry The Definitions We begin with a triangle labeled in the usual manner. hypotenuse θ adjacent opposite Figure - Standard Right Triangle opposite adjacent opposite hypotenuse Then we have: sinθ =, cosθ =, tanθ =, cscθ =, hypotenuse hypotenuse adjacent opposite hypotenuse adjacent secθ =, cotθ =. adjacent opposite Use these definitions, along with your special right triangles (and/or unit circle) to find sines, cosines, and tangents of the special angles. CAIC NOTES 09, PAGE OF 11
Some Identities Reciprocal 1 sinθ =, cscθ Quotient sinθ tanθ cosθ =, 1 cosθ =, secθ cosθ cotθ = sinθ 1 cotθ = tanθ Pythagorean sin θ cos θ 1 + =, 1+ cot θ = csc θ, tan θ + 1 = sec θ A Note about Squaring ( ) ( ) sin θ = sinθ sin θ [4.] Find the exact values of all six trig functions of the angle θ given in the figure. 8 1 θ Notice that you must use the Pythagorean Theorem to find the length of the third side: x + 8 = 1 x + 64 = 144 x = 80 x = 80. 8 So sinθ = =, 1 80 cosθ =, 1 8 tanθ =, 80 1 1 cscθ = =, secθ =, 8 80 80 cotθ =. 8 [.] Sketch a right triangle where tanθ =. 4 Well, since we know the lengths of two sides, we automatically know the third not that we really need it, in order to draw the triangle. 4 θ CAIC NOTES 09, PAGE 4 OF 11
[6.] If tanθ =, then find the exact value of sinθ. 4 Now we do need to know the length of the hypotenuse! + 4 = x 9 + 16 = x = x x =. Don't forget that is the length of the side opposite, and 4 is the length of the side adjacent. sinθ =. 9.4 Trigonometric Functions of Any Angle The Definitions Consider a point (x, y) out in the coordinate plane. Connect that point to the x-axis with a vertical line, and connect the point to the origin you now have a triangle! Thus, any point in the coordinate plane defines a triangle (well, almost any point ), and you can then define the six trig functions in terms of that point. Note that r is distance from the point to the origin, and θ is the counter-clockwise angle between the positive x-axis and the line connecting the point to the origin. y x y r r x sinθ =, cosθ =, tanθ =, cscθ =, secθ =, cotθ =. r r x y x y Notice that this definition means that lots of angles (an infinite number) all have the same sine and cosine values because the angle can wrap around the origin (angles of greater than 60º are allowed). Reference Angles Some people have a problem referring to the angle θ in the way that I did in particular, this allows angles with measures greater than 180º. Thus, some people talk about reference angles the smallest angle between the line connecting the point and the origin, and the x-axis (positive or negative). The Unit Circle As mentioned previously, every angle crosses the unit circle at a point (x, y). The radius (distance from that point to the center) is 1, so we can rewrite the earlier definitions y 1 1 x sinθ = y, cosθ = x, tanθ =, cscθ =, secθ =, cotθ =. Thus, we just need to know the x y x y coordinates of the point for each angle, and we can use those coordinates to find the values of all six trigonometric functions. See? The unit circle is just a memorization tool! [7.] Find the value of all six trigonometric functions for the angle determined by the point (, ). First of all, find r: + = r 4 + = r 9 = r r = 9. sinθ =, 9 cosθ =, 9 tanθ =, 9 9 cscθ =, secθ =, cotθ =. CAIC NOTES 09, PAGE OF 11
[8.] Evaluate cos 40º without a calculator. Notice that 40º = 60º + 4º. Thus, 40º and 4º point in the same direction rays along those 1 angles pass through the same points. Thus, cos 40º = cos 4º =. 9. Graphs of Sine and Cosine Functions First up: when we start graphing trigonometric functions, we MUST use radians. Hint: if you are sketching these functions without a calculator, then use a few choice points xintercepts, maxima and minima. You should know those points from earlier memorization! The Basic Graphs The horizontal scale on these graphs is 4 π, and the vertical scale is 1. y = sin x y = cos x Transformations y = a b x c + d and sin ( ) Naturally, you can transform these graphs. The templates are ( ) y a cos( b( x c) ) d = +. The parameters a, b, c and d all do exactly what they have always done but in trigonometry, they sometimes get special names. For example, the amplitude of the function is a, and the phase shift is c. Remember that the trigonometric functions are periodic they repeat a pattern every so often. Sine and cosine normally have a period of π; when transformed, this changes (don't forget that the period is a horizontal property!). CAIC NOTES 09, PAGE 6 OF 11
[9.] Find the period and amplitude of y = -cos (4x + 1). The amplitude is - =. The new period is the original period (π) divided by b (4): π π =. 4 [10.] Describe a sequence of transformations that changes the graph of y = sin x into y = sin (x + π) 1. Multiply the y-values by ; shift π units to the left; shift down 1 unit. 9.6 Graphs of Other Trigonometric Functions Sine and Cosine are continuous no gaps or breaks. The other trigonometric functions are not so nice. Here are their graphs: The horizontal scale on these graphs is 4 π, and the vertical scale is 1. y = csc x y = sec x y = tan x CAIC NOTES 09, PAGE 7 OF 11
y = cot x Of course, you can transform each of these the templates all look the same. One difference is that a is not called the amplitude for these graphs. If you need to sketch these graphs without a calculator, then you should begin with the asymptotes and x-intercepts/extrema. These should be fairly easy to find if you don't have them memorized If you need to graph these with a calculator, then you need to know the Quotient identities since calculators only have Sine, Cosine and Tangent available. [11.] Sketch the graph of y = cot x. Enter y ( 1/ tan x) = into your calculator CAIC NOTES 09, PAGE 8 OF 11
[1.] Solve (graphically) sec x = for x [ π, π ]. Enter y = 1/ cos x and y = -, then find the intersection. The solutions are -4.1888, -.0944,.0944, and 4.1888. 9.7 Inverse Trigonometric Functions The Need We often need to "undo" a function take what is usually the output, and make it the input, in order to find what is usually given (whew!). We use inverse functions for this. We now have six (well, three, at least) brand-new functions what about inverse functions? The Problem As we learned (reviewed) in Chapter, not every function has an inverse function. In particular, a function must be one-to-one (pass the horizontal line test) in order to have an inverse function. Unfortunately, none of the trigonometric functions is one-to-one! The Solution Something that we mentioned only in passing back in Chapter was that you can force two things to be inverse functions if you restrict the domain of one or both functions. And that is precisely what we must do to the trigonometric functions! Definitions and Limitations 1 The inverse function of y = sin x is y = arcsin x = sin x. Since sine is a function, the "exponent" of -1 really isn't an exponent it denotes the inverse. Some people get around this by CAIC NOTES 09, PAGE 9 OF 11
calling the function arcsine. You may (at some point) be told that there is a difference between arcsine and Arcsine. There is but the difference is not terribly important for most of us, and we won't linger on them. We will be using the lowercase version exclusively. π π The restricted domain of sine is,, which still produces a range of [-1, 1]. That means π π that the domain of arcsine is [-1, 1], and its range is,. Inverse Function Domain Range arcsine [-1, 1] π π, arccosine [-1, 1] [0, π] arctangent R π π, Very few (if any) people refer to inverse functions for secant, cosecant and cotangent so we won't, either. [1.] arcos(½) =? π 1 Since cos =, 1 arccos = π. [14.] arcsin(-½)=? 11π 1 Careful! You may be tempted to say that since sin =, the answer must be 11 but 6 6π π π that can't be; the range of arcsine is,, and 11 π isn't in that interval! The answer will be 6 an angle that is coterminal with 11 π π, but in the interval, 6π. That angle is π. 6 π [1.] arctan tan 4 =? π Since they are inverses, they just wipe each other out the answer is. 4 4π [16.] arccos cos =? Alas, 4 π isn't in the range of arccosine, so that can't be the answer! We need an angle that has the same cosine value as 4 π π, but in the range of arccosine ( [0, π] ). That angle is. CAIC NOTES 09, PAGE 10 OF 11
[17.] cos arcsin 1 =? This requires a specific technique, since isn't a special fraction from the unit circle / special 1 right triangles. remember that the arc functions take in real numbers, and spit out angles. Thus, what we ultimately want to do is take the cosine of some angle if only we knew what the angle was! We can draw it let arcsin = θ. 1 1 θ Now the question asks for cos arcsin, which is the same as cos θ can you find cos θ? I 1 1 should hope so! cos arcsin =. 1 1 CAIC NOTES 09, PAGE 11 OF 11