TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles

Transcription

1 TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ )

2 . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula measuement but also fo modeling peiodic phenomena, we must give a definition of an angle that is somewhat moe geneal than a vetex angle of a tiangle. If A and B ae distinct points, the potion of the staight line that stats at A and continues indefinitely though B is called a ay with endpoint A (Figue.a). An angle is detemined by otating a ay about its endpoint. This otation can be indicated by a cuved aow as in Figue b. The endpoint of the otated ay is called the vetex of the Figue. (a) (b) A. end point B. vetex α γ Teminal initial angle. The position of the ay befoe the otation is called the initial of the angle, and the position of the ay afte the otation is called the teminal. Angles will often be denoted by small Geek lettes, such as angle α in Figue b ( α is the Geek lette alpha). Angles detemined by a clockwise otation ae said to be positive and angles detemined by a clockwise otation ae said to be negative. In Figue.2a, angle β is positive ( β is the Geek lette beta); in Figue.2b, angle γ is negative ( γ is the Geek lette gamma). Figue.2 (a) (b) β is positive (counteclockwise) initial γ is negative (clockwise) teminal β initial γ teminal vetex vetex

3 One full tun of a ay about its endpoint is called one evolution (Figue.3a); one-half of a evolution is called a staight angle (Figue.3b); and one-quate of a evolution is called a ight angle (Figue.3c). The initial and teminal s of a ight angle ae pependicula; hence, in dawing a ight angle, we often eplace the usual cuved aow by two pependicula line segments (Figue.3c). Figue.3 (a) (b) one evolution staight angle. vetex Initial Teminal Teminal. vetex Initial (c) ight angle Initial Teminal vetex Angles that have the same initial s and the same teminal s ae called coteminal angles. Fo instance, Figue. 4 shows thee coteminal angles α, β and γ. Because diffeent angles can be coteminal, an angle is not completely detemined meely by specifying its initial and teminal s. Nevetheless, angles ae often named by using thee lettes such as ABC to denote a point A on the initial, the vetex B, and a point C on the teminal (Figue.5). Usually, the subtended angle is the smallest oataion about the vetex that caies the initial aound to the teminal. Sometimes, when it is pefectly clea which angle is intended, we efe to the angle by simply naming its vetex. Fo instance, the angle in Figue.5 could be called angle B. 2

4 Figue.4 Figue.5 teminal α β γ teminal C.. angle ABC A initial initial B Thee ae vaious ways to assign numeical measues to angles, Of these, the most familia is the degee measue: One degee ( ) is the measue of an angle fomed by 360 of a counteclockwise evolution. Negative angles ae measued by negative numbes of degees: fo instance, 360 is the measue of one clockwise evolution. A positive ight angle is of a clockwise evolution, and it theefoe has a measue of 4 80 ( 360 ) = 90. Similaly, 80 epesents o of a clockwise evolution What is the degee measue of one-eighth of a clockwise evolution? ( -360 ) = Example What faction of a evolution is the angle of 30? = 2 of a counte clockwise evolution. Although factions of a degee can be expessed as decimals, such factions ae sometimes given in minutes and seconds. One minute ( ) is defined to be of a 60 degee and one second ( ) is defined to be of a minute. Hence, one second is of a degee. Using the elationships = 60 o, = 3600 o, = 60, ' = 60 3

5 you can convet fom degees, minutes, and seconds to decimals and vice vesa. Although the degee measue of angles is used in most elementay applications of tigonomety, moe advances applications (especially those that involve calculus) equie adian measue. One adian is the measue of an angle that has its vetex at the cente of a cicle (that is, a cental angle) and intecepts an ac on the cicle equal in length to the adius (Figue.6). A cental angle of 2 adians in a cicle of adius intecepts an ac of length 2 on the 3 3 cicle, a cental angle of adian intecepts an ac of length, and so foth. Moe 4 4 geneally, if a cental angle AOB of θ adians (θ is the Geek lette theta) intecepts an ac AB of length s on a cicle of adius (Figue.7), then we have s = θ. It follows that AOB has adian measue θ given by the fomula s θ =. Figue.6 Figue.7 one adian ac length = B s = θ cente cente θ adians A Find the length s of the ac intecepted on a cicle of adius = 3 metes by a cental angle whose measue is θ = 4.75 adians. s = θ = 3 ( 4.75) = 4.25 metes. 4

6 A cental angle in a cicle of adius 27 inches intecepts an ac of length 9 inches. Find the measue θ of the angle in adians. Hee s = 9 inches, = 27 inches, and θ = s = 27 9 = 3 adian. Anothe elated fomula is that fo aea of a secto cut off fom a cicle of adius by a cental angle of adian measue θ. Secto θ adians Note that the aea A of this secto of the whole cicle as θ is to 2. That is, A θ 2 = 2 Thus, A = 2 2 θ A cental angle of 360 coesponds to one evolution; hence it intecepts an ac s = 2 equal to the entie cicumfeence of the cicle (Figue 8). Theefoe, if θ is the adian measue of the 360 angle, θ = s = 2 = 2 adians; that is, 360 = 2 adians, o 80 = adians. You can use this elationship to convet degees into adians and vice vesa. In paticula, = adian 80 and 80 adian = ( ). Thus, we have the following convesion ules: (i) Multiply degees by 80 to convet to adians. (ii) Multiply adians by 80 to convet to degees. 5

7 When no unit of angula measue is indicated, it is always undestood that adian measue is intended. Table. Degees Radians Application: Angula Velocity A fomula closely elated to the ac length fomula s = θ is the fomula which connects the speed (velocity) of a point on the im of a wheel of adius with the angula velocity ω at which the wheel is tuning. Hee, ω is measued in adians pe unit time. Example Detemine the angula velocity in adians pe second of a bicycle wheel of adius 6 inches if the bicycle is being idden down a oad at 30 miles pe hou. To solve, we must use consistent units v = ω 30mi h mi = 30 h 5280 ft mi h 3600 s = 44 s ft and the adius of the wheel is ft. Thus, 44 = ω o ω = (44) = 33 adians pe second

8 Section Poblems Find the angles A, B, C, D, E, and F in the figue shown at the fa ight. 2. Find the distance, x. 3. Find the angles A and B. Assume that the beam is of unifom width. 4. On a cetain day, the angle of elevation of the sun is o Find the angles A, B, C, and D that a ay of sunlight makes with a hoizontal sheet of glass. Assume the glass to be so thin as to not bend the ay. 5. Thee paallel steel cables hang fom a gide to the deck of a bidge. Find the distance x. 7

9 6. A dead fly is stuck to a belt that passes ove two pulleys 6 inches and 8 inches in adius, as shown in the figue. Assuming no slippage, how fast is the fly moving when the lage pulley tuns at 20 evolutions pe minute? 7. How fast (in evolutions pe minute) is the smalle wheel in poblem 49 tuning? 8. Assume that the eath is a sphee of adius 3960 miles. How fast (in miles pe hou) is a point on the equato moving as a esult of the eath s otation about its axis? 9. A nautical mile is the length of (ecall minute = of a degee) of ac on the 60 equato of the eath. How many miles ae thee in a nautical mile? 0. Find the volume of the boed hexagonal cylinde shown in the figue. Hint: Recall that the volume of any cylinde is detemined by the fomula: Volume = (Aea of Base) x (Height). Note also that if it wee not fo the cicula hole in the hexagonal base then the base could be patitioned into six conguent tiangles. 8