There are three standard ways of measuring angles: degrees, radians and grads.
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1 CHAPTER Trigonometry Section. Angles and Their Measures Degrees and Radians (and Gradians) There are three standard ways of measuring angles: degrees, radians and grads. Degree measurement is based on dividing the central angle of a circle in 0 equal arts. Radian measurement is based on dividing the central angle of a circle in equal arts. Gradian measurement is based on dividing the central angle of a circle in 400 equal arts. It is not commonly used and will be ignored here. Because of these definitions, we have the following relationshi. radians = 0 () = 0 = 0 Eamle.: Convert into degrees. Multilying by 0 gives 0 =. Eamle.: Convert 0 into radians. Multilying by 0 gives 0 0 = 9. Circular Sector A circular sector is a ortion of a circle, with the center of the circle as the verte of the angle q. r s q r The length of the arc s is given by s = rq, where q is measured in radians. The area A of the sector is given by A = r q, where q is measured in radians. Eamle.: A car travels along a circular arc with radius 40 feet and angle. How far did the car travel? Since the angle q must be in radians, we have s = feet 80 Eamle.4: A lawn srinkler srays along a circular arc with radius 4 feet and angle. What area is covered by the srinkler? Since the angle q must be in radians, we have Angular and Linear Velocity A = (4) 94 feet 80 For the uroses of this class, linear velocity is the same as seed. Using the standard method from beginning algebra, linear velocity = distance time Similarly, angular velocity is defined as =) v = s t The letter w is lower case omega. angular velocity = angle time =) w = q t Eamle.: Find the angular velocity of the minute hand of a clock. The minute hand of a clock makes a comlete revolution in 0 minutes, therefore w = 0 rad /min = 0 rad /min
2 Eamle.: The head of a foot long golf club rotates through an angle of 00 in 0. sec. Find the seed of the head of the golf club. The radius r of the circular ath is ft. The angular velocity is w = 00 (/80) 0.. rad/sec. Therefore, the seed of the club is v = rw = (.) 7 feet/sec 7 miles/hour Eamle.7: A car travels at 0 mh three-fourths the way around a circular ath. If the radius of the ath is 0 feet, find the angular velocity of the car. Since w = q t, and q = s r, w = q t = s/r t = r s t = v r Converting the seed 0 mh to 88 feet/sec, we have Section. Si Trigonometric Functions w = v r = rad/sec Right Triangle Trigonometry hyotenuse oosite side A adjacent side cos A = adjacent hyotenuse sec A = hyotenuse adjacent sin A = oosite hyotenuse csc A = hyotenuse oosite tan A = oosite adjacent cot A = adjacent oosite This section involves using trigonometry to solve word roblems. Two terms used in the homework are angle of elevation and angle of deression. The angle of elevation is an angle that starts at a horizontal line and oens uward. Similarly, the angle of deression is an angle that starts at a horizontal line and oens downward. Eamle.8: The angle of elevation from your eye to the to of a ainting is. The angle of elevation from your eye to the bottom of the ainting is 8. If you are standing feet from the wall holding the ainting, how tall is the ainting? y 8 When looking at the bottom of the ainting, we have tan 8 = =) = tan 8 When looking at the to of the ainting, we have tan = + y =) + y = tan Therefore, the height of the ainting is y = tan tan 8.8 feet Section. The Unit Circle An angle in standard osition is two rays, etending from the origin with the initial side of the angle along the ositive -aes. The terminal side of an angle is the other ray of the angle. terminal side reference angle A reference angle is the angle between the terminal side of the given angle and -ais.
3 Eamle.9: Find the reference angle for. Drawing the angle in the second quadrant leaves gives a reference angle of 80 = 0. A coterminal angle with a given angle is an angle whose terminal side is at the same location. Eamle.: Find two angles coterminal with. Drawing the angle in the third quadrant, we find the reference angle. angle = reference angle = 0 coterminal angle = Secial Angles Since the reference angle is 0, we have a coterminal angle of. In general, coterminal angles can be found by adding multiles of 0. Therefore, + 0 = 70 is another coterminal angle sin 0 cos 0 tan 0 undefined Eamle.: Find the eact value of sin and cos. Start by drawing the angle and noticing the angle is in the second quadrant and has a reference angle of. From the secial angles, we know that sin = cos =. and Since the sine function is ositive in the second quadrant and the cosine function is negative in the second quadrant, we have sin = and cos = Section.4 Grahs of Sines and Cosines This section focusses on the grahs of the sine and cosine functions. y = A cos(b + C)+D and y = A sin(b + C)+D For both of these functions, amlitude = A eriod = B hase shift = C B vertical shift = D Eamle.: Find all the imortant information for f () = sin + + and sketch the grah.
4 Alying the above concets to the given equation, we have the following figure. 7 Figure.: Grah of f () = sin + + using ale ale, ale y ale The eriod is found by using eriod = B = = and the hase shift is found from hase shift = C B = / = Reading directly from the function, we have an amlitude of and a vertical shift of. Periodic behavior Periodic behavior is a attern that reeats itself. For eamle, the temerature in a city is eriodic, both on an annual scale (cold every winter, warm every summer) and on a daily scale (cooler at night, warmer in the day.) Certain tyes of eriodic behavior can be modeled by a sine or cosine function. This tye of modeling can be done without using a calculator. Eamle.: The daily summer temerature in Fremont can be modeled by a sine function. If at :00 AM the temerature reaches is coldest value of 48 F and at :00 PM, the temerature reaches it highest value of 78 F, find a cosine function to model the temerature in the room. Let t = 0 reresent midnight. Since the temerature varies from 48 to 78, these reresent the high and low oints on the sine grah. Also, we know that the low temerature occurs at time t =. This information is summarized in Figure.. 80 Temerature (in F) Time (in hours after midnight) Figure.: Time versus Temerature With this grah, we can now generate the constants in the equation T(t) =A cos(bt + C)+D The total height of the wave is = 0 so the amlitude of the wave is. Since the grah is an inverted cosine wave, we have A =. Since the vertical center of the wave is T =, the vertical shift is given by D =. The temerature has a eriod of 4 hours. Therefore, eriod = B = 4 =) B = Since the hase shift is t =, we can set determine value of C by substituting known values into B + C = 0. ()+C = 0 =) C = Therefore, the equation is given by T(t) = cos t +
5 Section. Grahs of Other Trig Functions The four functions tangent, cotangent, secant and cosecant can all be written in terms of sine and cosine. tan A = sin A cos A cot A = cos A sin A sec A = cos A csc A = sin A Since each of these four functions has a denominator that becomes zero for certain values of A, each of the functions must contain vertical asymtotes. In all cases, the vertical asymtotes occur when the denominator has a value of zero. Eamle.4: Use the grah of y = cos and grah of y = tan. Start by sketching the grah of f () =cos to determine the asymtotes. 4 Figure.: Grah of f () =cos( ) using ale ale, ale y ale tiongrah of f () =cos using ale ale, ale y ale At each lace the grah of cosine crosses the -ais, draw a vertical line for the asymtotes of tangent grah. 4 Figure.4: Asymtotes of f () =tan( ) using ale ale 4 Figure.: Grah of f () =tan( ) using 4 ale ale 4 Figure.: Grah of f () =tan( ) using 4 ale ale
6 Section. Inverse Trig Functions The inverse trigonometric functions are written in two forms. For eamle, the inverse function for f () =sin is written as either f () =arcsin or f () =sin. Note: sin = (sin ) Because of the misleading notation, I much refer the arcsin notation. Since none of the trig functions are one-to-one, to find the inverses, we must use a restricted domain. The effect of this is to restrict the range of the inverse trig function. Eamle.: Restrict the domain of y = sin so that the resulting grah shows a one-to-one function. Below is the grah of f () =sin on the interval ale ale drawn as a dashed curve. To have a one-to-one function, use only the ortion of the grah in the interval ale ale. Figure.7: Grah of f () =sin Eamle.: Sketch a grah of y = arcsin and use the grah to state the range. The grah of an inverse function y = f () can be found by switching the and y value of y = f (). We also know the range of f () is the same as the domain of f (). Therefore, the range of the inverse sine function is the same as the domain of the the restricted sine function. Below is the grah of y = arcsin on the interval ale y ale drawn as a dashed curve. To ensure that arcsine is a function, use only the ortion of the grah in the interval ale y ale Figure.8: Grah of f () =arcsin For the reasons outlined in the revious eamles, the range of each of the inverse trig functions is restricted to a secific interval. The three most common are given below. function range arccos arcsin arctan 0 ale y ale ale y ale < y < One of the tyes of roblems in which you will see the inverse trig functions is demonstrated in the net eamle. Eamle.7: Write tan arccos without trig functions.
7 Start by searating the roblem into two arts by making the substitution q = arccos. The question can then be restated as Find tan q if q = arccos From q = arccos, we have cos q = q From this triangle, we can use the Pythagorean Theorem to find the length of the vertical segment (call it y). + y = =) y = 4 4 q From this new triangle, we can now write tan arccos 4 = tan q =
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