P.7 Trigonometry. What s round and can cause major headaches? The Unit Circle.

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1 P.7 Tigonomet What s ound and can cause majo headaches? The Unit Cicle. The Unit Cicle will onl cause ou headaches if ou don t know it. Using the Unit Cicle in Calculus is equivalent to using ou multiplication facts in Algeba. In case ou ve fogotten what it looks like, hee it is. Page of 5

2 The odeed pais on the Unit Cicle give the cosine and sine values of some ve useful angles in adians,, cos,sin. Just knowing the cosine and sine values of these angles, with the help of such that the Recipocal and Ratio Identities, ou will be able to also find the othe fou tigonometic values, namel tangent, cotangent, secant, and cosecant. sec cos csc sin sin tan cos cos cot sin But what eactl is a tig atio anwa? You should neve foget to emembe that a tig atio is nothing moe than a unitless atio of two of the thee sides of a ight tiangle whose efeence angle is detemined b the independent angle. The definition of the si tig atios in tems of,, and (which will wok fo a cicle of an size, not just the unit cicle when ) cos sec sin csc tan cot Regadless of the size of the cicle, these atios, fo given angles, will alwas simplif to the same value. This is because the scale facto alwas divides out (as well as an units). Test out ou unit cicle memo b quickl giving the simplified, eact value of the following: a) 5 sin 3 b) 5 cos 6 c) 7 tan d) 3 cot e) 3 sec f) csc Page of 5

3 You should have coectl obtained the following: d) a) cot sin b) cos c) tan e) sec f) csc DNE o Does Not Eist If ou have a headache at this point, ou should (e)memoize the Unit Cicle and begin quizzing and equizzing ouself until ou ve masteed this. Sometimes, though, we ll be woking with angles and atios not on the unit cicle, and we ll might not even have ou calculato. Eample : 3 Find all the emaining five tig values of if sin and cot 0, Then find such that 0. What is the efeence angle? What othe angle 0 has the same sine value as? It s woth noting hee the diffeence between an invese tig opeation, vesus and invese tig function. When solving fo an angle, we take the invese (o ac) tig function of both sides of the equation. We must then find all the solutions in a specified inteval, such that the tig function of each angle is the oiginal atio. Fo eample... Page 3 of 5

4 Eample : Fo (a) 0 and (b) all eall numbes, solve sin In calculus, if the invese o ac tig equation is eplicitl given to us in the poblem, we will assume it is talking about the invese tig function. That means fo an input (a atio) thee can onl be one output. Hee s a simila tpe of question pesented slightl diffeentl. Eample 3: Find an algebaic epession fo tan acsint. Anothe impotant tigonometic concept integal fo calculus success is the abilit to use tig identities to simplif epessions. We ve alead see the Recipocal and Ratio Identities, but thee ae a handful moe that show up fom time to time. As a geneal ule thoughout calculus, we will want to simplif as eal as we can in a poblem, and as often as we can as we wok though the poblem. Things we can do ae to epand epessions, combine like tems, divide out common factos, simplif eponents and log epessions, and finall, make tigonometic identit substitutions. Remembe: Simplif Eal And Often! The Pthagoean Identities (PIDs): Papa PID Bab PID Bab PID cos sin tan sec cot csc These identities can be algebaicall manipulated to ceate othe equivalent foms, fo eample: sec tan Page of 5

5 The Double Angle Identities: Single Tem sin sin cos Diffeence of Squaes cos cos sin These identities allow ou to epess an angle in tems of an angle half its size. Late in ou studies, these identities will be useful to ewite in integal epession into something moe easil ecognized. You ll see. The Powe-Reducing Identities: Single Tem Diffeence of Squaes sin cos cos cos The identities will also be invaluable when we stud integation, as the allow us to go fom staing at something that s impossible to do, to something that s as eas as taking cand awa fom a bab who s ting to give awa his cand. That s basicall it. Know ou identities, know the Unit Cicle, and don t mess up. Hee a cute Tedd Bea, pool outlined, with a simple message fo ou. Let s get on to calculus. Page 5 of 5