Angles and Their Measure; Basic Definitions

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1 Angles and Their Measure; Basic Definitions This section introduces us to many of the definitions that we will use in our study of trigonometry. Most definitions are fairly intuitive and you may already know many of them anyway. To summarize the major ideas:. An angle is formed by two rays meeting at a common point. For convenience, we fix the point at the origin of the Cartesian plane.. The starting side of the angle is the initial side, usually placed on the positive x- axis, and the ending side of the angle is the terminal side. 3. A positive angle has counterclockwise rotation from initial to terminal sides. A negative angle has clockwise rotation. 4. Coterminal angles have the same initial and terminal sides, but may have different numeric measurements (for example, going around a complete revolution and finishing at the same terminal ray). 5. Degree measurement is where the circle is divided into 360 degrees, with right angles being 90 degrees and straight lines being 80 degrees. 6. The four quadrants are in order: Q: upper right quarter, Q: upper left quarter, Q3: lower left quarter, Q4: lower right quarter. 7. A right angle has a measurement of 90 degrees. Above 90 degrees, the angle is obtuse. Below 90 degrees, the angle is acute. 8. If two positive angles sum to 90 degrees, the angles are said to be complementary. 9. If two positive angles sum to 80 degrees (a straight angle), the angles are said to be supplementary. Radian measurement is a practical method of measuring angles. The idea is this: A circle with radius unit is drawn. By simple geometry, we know the circumference of a circle is C = r. Since r =, the circumference is. In degree measurement, one complete rotation is 360 degrees. So we say 360 degrees =. Therefore, dividing by, we get 80 degrees =. This leads to two conversion formulas:. degree = radian = degrees. (This is roughly 57.3 degrees) You will go far if you memorize some basic measurements: 30 degrees = 6 45 degrees = 4 60 degrees = 3 90 degrees = 80 degrees = 360 degrees =.

2 As with any new form of measurement, you must get used to using it in practice. It may seem awkward at first but it will become intuitive and second-hand in time. The beauty of radian measurement is that it is independent of the size of the circles or triangles used to create angles. An immediate application of angles and measurement is the formula s = rθ, where θ is the radian measure of an angle, r the radius of a circle, and s the arc length of the portion of the circle subtended by the angle. This formula relates the angular distance traveled s if the circle s radius r and angle measure θ are given. Right Triangle Trigonometry We now delve into the basic trigonometric functions pertaining to angle θ. Consider a right triangle. Angle θ is placed in one corner (not the right angle corner). Relative to θ, the triangle has an adjacent side, an opposite side, and a hypotenuse. From this, we define the sine of θ, written sin θ, to equal the ratio of the opposite side over the hypotenuse. The cosine of θ, written cos θ, is the ratio of the adjacent side over the hypotenuse. The sine and cosine are the two basic, axiomatic definitions. All other definitions can be derived from manipulations involving sine and cosine. The tangent of θ, written tan θ, is sin the opposite side over the adjacent side, equivalently, tan θ = cosθ. The rest are: sec θ =, cotθ = =, csc θ =. cosθ tanθ cosθ sinθ sinθ Some measurements of common angles can be determined by special triangles. A triangle that is will have legs of equivalent measure. If each leg is unit, then the hypotenuse is units and we can then show that the sin 4 = =, (remember, = 4 o 45 ). Can you determine the cosine and tangent values for this same angle? Another common triangle is the triangle, which is half an equilateral triangle. Pythagoras Formula helps calculate the side lengths. You should be able to convert all of the six trigonometric functions back into forms involving sine or cosine only. Also of importance is the Pythagorean identity sin θ + cos θ =. This is true for any angle θ. Note: The expression sin θ is just shorthand for (sinθ ). If using your calculator, be sure to know when you are using degree measurement and radian measurement, and to convert your calculator into the appropriate mode when necessary. θ

3 Trigonometric Functions of Any Angle or Real Number We can extend the various trig functions so that the input is any real number, and hence we can treat these as functions in the usual sense. A function like = sin x now makes perfect sense; x is simply some input and sin acts upon it. Useful Fact: A handy mnemonic is ASTC, in which A represents all, S represents sine, T tangent and C cosine. The A goes into Quadrant I, S into Quadrant II, T into Quadrant III and C into Quadrant IV. Remember: All Students Take Calculus. This reminds us which of the trigonometric functions are positive in which quadrants. For angles greater than 90 degrees ( ), it is useful to consider reference angles, which are the angles made by the terminal side and the nearest portion of the x-axis. The table found here: shows the basic values of the sine, cosine and tangent for the most common angles. Know this table and don t forget it! Lastly, we note that since there is no limit as to how large the value of θ can get, the values of sin θ, cos θ, tan θ and the rest will repeat for ever, in a periodic fashion. The sine and cosine functions have period, since sinθ = sin( θ + n ), where n is an integer. Similarly for cosine. The tangent function has period. The expression n simply represents n full turns around the unit circle. Trigonometric functions also have symmetry. The cosine function is even, while the sine and tangent functions are odd. Instructor s Note: I very rarely deal directly with sec θ, cot θ and csc θ, choosing instead to rewrite these functions into forms involving sin θ, cos θ or tan θ. You will find that this will be just as easy and less to remember. Graphs of Sine and Cosine Functions We will learn how to graph the sine and cosine curves in this section. We first start with the basic curves = sin x and g( x) = cos x. Here s a handy practice and reference link: Note the period. Each period represents one cycle. We will now apply various tricks with our graphs. Suppose we multiply = sin x by, getting = sin x. The acts as a vertical stretch of sin x. In general, if we are given = a sin x, the absolute value of a is the amplitude of the graph. The same applies for the cosine graph.

4 If we multiply the angle by a coefficient first, we affect the period of the graph. For example, if we have = sin x, the graph actually shrinks in the horizontal direction. The period is no longer. In this case, the period is. In general, given = sin bx, the period is p =. Similarly for cosine. b Given = sin x, we can consider = sin( x c). The c acts as a horizontal shift, shifting the graph c units to the right. Again, same for cosine. Lastly, if we add a constant to the whole equation, we achieve a vertical shift. For example, = 3 + sin x is just the sine graph moved three units up. We can summarize these stretches and shifts into one all-encompassing equation: = a sin[ b( x c)] + d (again, similarly for the cosine). For example, if we are given = 3sin(x + 4) + 6, we first factor out the inside the sine, getting = 3sin[( x + )] + 6. We conclude the amplitude is 3, the period is (since b = and p = b = = ), the horizontal shift is two units to the left (why?), and the vertical shift is six units up. Note: if you are given = sin x + 3, this means the x is inside the sin, and the 3 is outside. On the other hand, if we wanted to group everything inside the sine function, we d write = sin(x + 3). Use parentheses to be clear what is being acted upon by the sine or cosine. The link has lots of good examples. Graphs of Other Trigonometric Functions We will be graphing the functions = tan x, = cot x, = sec x and = csc x. To understand these graphs, it is imperative that you know the basic graphs of = sin x and = cos x. Most important, we need to remember these two points:. sin x = 0 when x = n, where n is any integer.. cos x = 0 when x =, where n is an odd integer. For tan x, we use the fact that n sin x tan x =. cos x

5 Since cos x = 0 at PreCalculus Generic Notes x = for odd n, tan x will have a vertical asymptote at each of these n points. Also, since sin x = 0 at n for integer n, tan x will have a root (cross the x-axis) at each of these points. All you need to do is graph one branch of the function; it repeats itself forever. For = cot x we use the fact that cos x cot x = =. sin x tan x We locate vertical asymptotes and x-intercepts as we did for the tan function. Again, note the repeated, periodic behavior of the graph. One branch is graphed, and it repeats forever. The graphs of = sec x and = csc x are called reciprocal functions since sec x = and cos x csc x =. sin x Note the presence of asymptotes, and also note that = sec x and = csc x have no roots! Can you explain why? Does this violate the fundamental theorem of algebra? Inverse Trigonometric Functions We will need to use inverse functions of the three main trigonometric functions, so now we will define them. Consider the graph of = sin x. It is obvious that it does not pass the horizontal line test and is not one-to-one. To remedy this, we take just a portion of the graph, just enough to account for all the values in the range once and once only. If we restrict = sin x for x, we now have a one-to-one graph. We now define the inverse of this graph to be the inverse sine of x, written = arcsin x or sin x. It s important to note that given = arcsin x, the x is not an angle, while y is an angle. For instance, if you saw = arcsin 0. 5, you would say What is the angle y such that when I sine it, I get 0.5? Please note the domain and range of the arcsin function. Two more common inverse functions are = arccos x and = arctan x. The arccos function is developed in the same manner as the arcsin function (restrict 0 x ), while the arctan function is one-to-one on one branch, so by ignoring all the

6 other branches, it s easy to form the arctan function. Please note the domains and ranges of each, given with each graph. Applications and Models This section is devoted to some of the applications (there will be more!) of the trigonometric functions. Generally, trigonometry is useful for problems involving angles, and also for any periodic phenomenon, such as the temperatures throughout the year. Look over your text for good ideas of how to solve for various angles and/or lengths. You definitely want to make a sketch and label as much as you can. Using the Pythagorean formula and basic geometric concepts, you should be able to solve for the remaining side and/or angle. Don t be shy about using your calculator. Note that these problems tend to use degree measurement, so be sure that you are in the right mode. Please work these three examples by yourself and then check your answer against the book s solution. A knowledge of angles, trigonometry and geometry is extremely useful for navigation by compass. You can use your compass to find the bearing of an obvious object (a hilltop, for example), relative to magnetic north, and by repeating this process, you can trace your place relative to the terrain. Harmonic motion is a term used for any phenomenon that repeats itself on a regular basis for example, a bobbing spring, ripples in water, and a plucked string. A bobbing object can be modeled using a sinusoidal curve. (The term sinusoidal is used to represent any sine or cosine graph.)

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