Unit 2 - The Trigonometric Functions - Classwork

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Unit 2 - The Trigonometric Functions - ClassworkGiven a right triangle with one of the angles named " , and the sides of the triangle relative to " namedopposite, adjacent, and hypotenuse (picture on the left), we define the 6 trig functions to be:The Basic Trig Definitions!!oppositehypotenusethe sine function :sin " the cosecant function : csc" hypotenuseoppositeadjacenthypotenusethe cosine function : cos" the secant function : sec" hypotenuseadjacentoppositeadjacentthe tangent function : tan " the cotangent function : cot" adjacentoppositeGiven a right triangle with one of the angles named " with " in standard position, and the sides of the trianglerelative to " named x, y, and r. (picture on the right), we define the 6 trig functions to be:!ysin!" !rxthe cosine function : cos" rythe tangent function : tan " xryrthe secant function : sec" xxthe cotangent function : cot" ythe sine function :!the cosecant function : csc" The Pythagorean theorem ties these variable together : x 2 y 2 r 2You MUST, MUST, MUST know the above thoroughly, inside and out, backwards and forward, and can neverforget it. It must be part of you. Expect quizzes every day for the immediate future to test whether you know!these definitions.You will find that if you learn them now, this section will be incredibly easy. If you learnthem and immediately forget them, you will struggle throughout this course.A good way to remember the basic definitions is to remember the terms SOH-CAH-TOA. Sine Opposite,Hypotenuse, Cosine Adjacent, Hypotenuse . Tangent Opposite, Adjacent. For the other trig functions(called the co-functions), Sine goes with Cosecant (S goes with C), Cosine goes with Secant (C goes with S),and the other functions both use the words tangent.Finally, remember that there is no such thing as sine. Sine doesn’t exist by itself. It is sin " or sin " or sin x .Every trig function is a function of an angle. The angle must be present.2. Basic Trigonometric Functions-1-www.mastermathmentor.com - Stu Schwartz!!!

Example 1) Let P be a point on the terminal side of " . Draw a picture and find the 6 trig functions of " .a) P (3,4)b) P (15,8)!!!!c) P (5,2)!d) P (1,7)!e) P (1,1)!e) P(2, 7)!Quadrant Angles:Let’s examine the trig functions if point P is not in the first quadrant. Let’s make a chart of the signs of x, y, andr in all of the quadrants and thus, the signs of the trig functions in those quadrants. (r is always positive)2. Basic Trigonometric Functions-2-www.mastermathmentor.com - Stu Schwartz

A good way to remember this is the term: A-S-T-C. It says the quadrants in which the 3 basic trig functions arepositive: (All – Sine – Tangent – Cosine)When we draw pictures of trig functions in quadrants other than quadrant I, the triangle is always drawn to thex-axis. The angle inside the triangle will be called the reference angle. It is defined at the acute angle formed bythe terminal side of " and the horizontal axis.Example 2) Let P be a point on the terminal side of " . Draw a picture showing the reference angle and find the6 trig functions of " .!a) P ("8,6)b) P (7,"24)!!!!(c) P ("2,"2)!d) P "1, 3)!Example 3) We can be given information about one trig function and ask about the others. Draw a picture.a) If sin " 122, " in quadrant I, find cos" and tan ". b) If cos " , " in quadrant IV, find sin" and tan ".133!!6d) If csc " , find cos" and cot" .5c) If tan " 3, " in quadrant III, find cos" and csc ".!!2. Basic Trigonometric Functions-3-www.mastermathmentor.com - Stu Schwartz

Example 4) In what quadrant(s) isa) sin " 0 and cos" 0!b) sec " 0 and cot" 0!c) csc " 0 and cos" 0!d) all trig functions are negative?!Trig functions of quadrant angles:The picture below shows quadrant angles: Choose a point for each quadrant angle, determine x, y, and r, anddetermine all six trig functions for those angles: Note that angles can be in degrees or in radians.0 (or 0)Point ( , )x y r sin0 csc0 cos0 sec0 tan 0 cot0 180 (or !)Point ( , )x y r sin180 csc180 cos180 sec180 tan180 cot180 # "&90 %or ( Point ( , ) 2'x y r sin90 csc90 cos90 sec90 tan 90 cot90 # 3! &270 % or( Point ( , )2' x y r sin270 csc270 cos270 sec270 tan 270 cot270 Example 5) Calculate the following without looking at the chart above:!a) 5sin 90 "12cos180 b)!!c) (4sin 90 " 2cos270 " 5)!26 tan180 3csc270 "2sec0 #"6sin " 5cot 90 2d)23cos2 0 " ( 3cos0)!2. Basic Trigonometric Functions-4-www.mastermathmentor.com - Stu Schwartz

Domain and Range of trig functions:yxand cot" , we have to worryxyabout angles where y 0 or x 0. x 0 along the y-axis so we cannot take the tangent of 90 or 270 . y 0along the x-axis so we cannot take the cotangent of 0 or 180 . For the csc function we have to be concernedabout angles where y 0 ( 0 or 180 . ) and for the sec function, we have to be concerned about angles where!x 0 ( 90 or 270 ).!!!Range: Since we know that trig functions are based on the picture below and that in any right triangle,!y smaller, we find that the range of ther x and r y , r must always be the larger side. So since sin " rlargerylargersine (and cosine) functions must be less than (or equal to) 1. And since csc " , the range of ther smallery! (or equal to) 1. Since tan " , we find that there is nocosecant (and secant) functions must be greater thanxrestriction on the values of the tangent function and cotangent functions. This can be summarized by the table!on the right:Domain: We can take the sine and cosine of any angle . But since tan " !Domain :Range :sin " : all real numbers-1 # y # 1 or [-1,1]cos" : all real numbers-1 # y # 1 or [-1,1]tan " : " 90 ,: " 270 all real numbers or (-%,%)csc " : " 0 ,: " 180 y # &1 or y ' 1 or (-%,-1] ( [1,%)sec " : " 90 ,: " 270 y # &1 or y ' 1 or (-%,-1] ( [1,%)cot " : " 0 ,: " 180 all real numbers or (-%,-1] ( [1,%)Special Triangles:! of sides in both 30 " 60 " 90 and 45 " 45 " 90 triangles.You must know the relationship!!In a 30 " 60 " 90 , the ratio of sides is 1" 3 " 2In a 45 " 45 " 90 , the ratio of sides is 1"1" 22. Basic Trigonometric Functions-5-!!www.mastermathmentor.com - Stu Schwartz!

So, complete the chart:"" !%30 or '# 6&!" !%45 or '# 4&sin "cos"!tan "!csc "!sec "!cot "!" !%60 or '# 3&The Special (Friendly) AnglesAny multiple of 30 ,45 or 60 is considered a special angle (or a quadrant angle) and we can compute trigfunctions of these angles. A) Draw it. B) Establish the quadrant and fill in the signs of the sides rememberingASTC. C) Find the reference angle (which will be 30 ,45 or 60 ) D) It is one of the special angles above.Determine the lengths of the sides (the signs are waiting for you). E) Find the trig functions of these angles.!RadiansDrawingReference sin ""cos"tan "csc "sec "cot "angle!120 !!!!!!135 150 210 225 240 300 315 330 2. Basic Trigonometric Functions-6-www.mastermathmentor.com - Stu Schwartz

Example 6) Calculate each of the following expressions. Do not look at the chart on the previous page as youwill not have it in an exam. As you did, draw a picture which will help you to calculate the values of trigexpressions. Label the picture in case you have to use it again.b) "2 tan 2 150 4cot 2 300 a) 8sin 30 " 6cos60 !!c) sin 2 315 cos2 315 d) ( 4 sin150 cos240 )!2!2 5"2 2" '# sinf) &2cot)%43("5sin90 " 2cos120 e)"5sin90 2cos120 !!Co-terminal Angles:So far, our angles have all been between 0 and 360 . What about angles outside that range? We will find thatsince 360 represents one full rotation, that when we take a trig function of an angle greater than 360 , thereference angle is the same as the angle created when subtracted 360 from the original angle. So we can makethis claim. We may add or subtract any multiple of 360 (2π) to any angle and the trig functions of that angle!remain the same. Not that we are not saying the angle remains the same; 100 and 460 are clearly different!!angles, but sin100 sin460 !!Example 7) For each angle given, find the angle between 0 and 360 which is co-terminal and then find the!signs of the trig functions of that angle.!"400 Co-terminalangle(between0 and 360 )!sin "!cos"!tan "!!csc "!sec "cot "!850 !1275 "231 "721 17"32. Basic Trigonometric Functions-7-www.mastermathmentor.com - Stu Schwartz

Unit 2 - The Trigonometric Functions - Homework1. Let P be a point on the terminal side of " . Draw a picture showing the reference angle and find the 6 trigfunctions of " .!a)!P (12,9)!b) P (30,16)!(c) P (1,2)d) P 3, 7!)!e) P ("8,"6)f) P (1,"3)!!(g) P 6," 13)!(h) P " 2," 2)!2. Given information about one trig function, find other trig functions:34, " in quadrant IV, find sin " and tan ".a) If tan" , " in quadrant I, find cos" and sin ". b) If cos" 23!!2. Basic Trigonometric Functions-8-www.mastermathmentor.com - Stu Schwartz

55c) If sin" , " in quadrant II, find sec " and cot ". d) If sec" # , " in quadrant III, find sin " and tan ".82!!e) If tan" #5, " in quadrant IV, find sin " and sec ". f) If cos" !2and sin" 0, find sin " and tan".3!6g) If sec" , find sin " and tan".5h) If tan" !4 5, find sin " and cos".5!3. In what quadrant isa) sin " 0 and cos" 0!c) sec " 0 and tan" 0!b) csc " 0 and cot" 0!d) csc " 0 and cos" 0!4. Find the value of the following (do not look at the chart – make a small picture and calculate the values)a) 5sin 90 " 7cos180 !2b) 4sec0 7csc270 !#&33" 3sec"(d) %6cot2 '!f) (sin 270 " sec0 )(sin 270 sec0 )!-9-2c) sin 180 cos 180 !e) cos0 sin 270 " cos270 sin 0 !2. Basic Trigonometric Functionswww.mastermathmentor.com - Stu Schwartz

5. For each statement, determine whether or not it is Possible (P) or Impossible (I).a) sin " #5c) 2cos" 5.5 4e) csc " sin # .5b) tan " 1 3.79d) sin " cot # 8f) sin " cos # 2!!6.Findthevalueofthefollowing(donotlookatthe chart – make a small picture and calculate the values)!!!!a) 6sin 30 " 4cos150 b) 8sin 60 " 4sin 300 !!c) (4 tan120 )(8cos225 )d) 6sin 315 8 tan135 !!e)8csc30 cot 330 f) "2cos225 " 4cot 315 3!!g) sin 2 225 " cos2 225 !h) cos3 630 " csc3 ("30) !2 "2" 'i) &sin # 4cos )3(% 6! 2 3"'42 7"# cscj) &cos)46(%!7. For each value of " , determine the co-terminal angle and the signs of the trig functions of that angle.Co-terminal"sin "cos"tan "csc "sec "cot "angle(between!0 and 360 )!!!!!!700 1525 "485 !2.5""20#72. Basic Trigonometric Functions- 10 -www.mastermathmentor.com - Stu Schwartz

the sine function: sin" y r the cosecant function: csc" r y the cosine function: cos" x r the secant function: sec" r x the tangent function:tan" y x the cotangent function:cot" x y The Pythagorean theorem ties these variable together: x2 y2 r2 You MUST, MUST, MUST know the above thoro

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