It s all about the unit circle (radius = 1), with the equation: x 2 + y 2 =1!

Transcription

1 Understing Trig Functions Preared by: Sa diyya Hendrickson Name: Date: It s all about the unit circle (radius = 1), with the equation: x + y =1! What are sine cosine values, anyway? Answer: If we go for a ride on the unit circle, starting at the oint (1, 0) l on some oint (x, y), the cosine value for the distance travelled is the x coordinate of the oint, while the sine value is the y coordinate of the oint! More secifically, we define sine, cosine tangent as follows: 1 cos : the x coordinate of the oint we l on (on the unit circle) after travelling the distance direction indicated by radians/degrees, from the oint (1, 0). sin : the y coordinate of the oint we l on (on the unit circle) after travelling a distance direction indicated by radians/degrees, from the oint (1, 0). tan : the ratio of sine cosine at, given by: tan = sin cos Notice that to evaluate tan, we simly need to evaluate sine cosine at! Making Math Possible 1 of 10 c Sa diyya Hendrickson

2 Secial Distances Points Level: x + y =1 To find sine cosine, we need to identify two things: 1. The distance direction that we are required to travel on the unit circle, given by radians/degrees. This distance direction can be found in the jaws of sine cosine.. The oint (x, y), on the unit circle, that we l on after travelling the required distance. There are infinitely many distances that we can travel to reach oints on the unit circle. However, the distances corresonding to oints that are easiest to determine receive secial attention. We will refer to them as stard distances stard oints. Distance on x axis Point Reached Diagram 0 rad = 0 (1, 0) rad = 180 ( 1, 0) rad = 0 (1, 0) Distance on y axis Point Reached Diagram rad = 90 (0, 1) rad = 70 (0, 1) Distance in Quadrant I Point Reached Diagram rad = 0, 1 rad = 5, rad = 0 1, Making Math Possible of 10 c Sa diyya Hendrickson

3 The Power of Symmetry Level: x + y =1 Recall that circles are symmetric. Symmetry allows us to use the stard distances oints in Quadrant I (QI) to easily determine stard distances oints in Quadrant II (QII), Quadrant III (QIII) Quadrant IV (QIV)! The comlete unit circle is given below. Notice that if we draw horizontal or vertical lines to connect oints that are across from each other, their coordinates are the same excet for the ossible di erence of a sign! This is the ower of symmetry! On the next age, we will take a closer look at how we can use this to our advantage. Making Math Possible of 10 c Sa diyya Hendrickson

4 Reference Angles Level: x + y =1 Reference angle: a reference angle (R.A.) is a stard distance/ radian inside of Quadrant I, namely: = 0, = 5 = 0 These angles are used to describe the shortest stard distance along the unit circle between a oint inside of a quadrant the x axis. This can be seen in the diagram below. *** Reference angles will hel us find sine cosine values! *** Exercise 1: Use the unit circle on Page to determine which radians have a R.A. of In other words, which radians are 0 from the nearest x axis?. Solution: Radians with R.A = are: 5, Now, let s make a coule of observations: 1 All of these radians have the same denominator of The oints associated with this reference angle have the form:, 1!,with either a lus sign or minus sign in the boxes. Making Math Possible of 10 c Sa diyya Hendrickson

5 Reference Angles Level: x + y =1 Exercise : Use the unit circle on Page to determine the radians the form of the oint associated with each reference angle. Solution: R.A. Associated Radians Point Form 5, 7, 5, , 1, 1, Is there a strategy for easily identifying which oint is associated with each R.A.? Answer: Yes! Let s take a moment to analyze our stard oints in Quadrant I, given by:, 1,, 1, Notice that: 1. The oints above consist of only three distinct coordinates! In other words, there are only stard sine cosine values in QI: root one over two, root two over two root three over two. 1 1 = < <. In QI : 1 is the smallest stard sine cosine value, while is the midsize stard value is the largest stard value. 1. The coordinates always come in a air. Therefore, whenever sine is 1, cosine must be by default ( vice versa)! On the other h, fills both coordinates of a oint!. Sine grows with the radians/distances in QI. Let s exlore this further on the next age! Making Math Possible 5 of 10 c Sa diyya Hendrickson

6 Reference Angles Level: x + y =1 Notice that as we travel u the unit circle in QI, achieving larger distances/radian measures, sine (the y coordinate on the unit circle) is also getting larger! Notice that our stard distances/r.a., in increasing order, are given by: smallest midsize largest (S) (M) (L) Stard Distances/R.A.: < < Stard Sine Values:, 1!,!, So, driving the smallest stard distance/r.a. of brings us to the smallest stard sine value of 1, while travelling the largest stard distance/r.a. of brings us to the largest stard value of! Moral of the Story: For each R.A, find the associated oint by first identifying the sine value, based on size! Then, fill in the cosine value by default! Exercise : Suose we are asked to evaluate cos 5 containing the x coordinate that we need?. What is the form of the oint Making Math Possible of 10 c Sa diyya Hendrickson

7 Reference Angles Level: x + y =1 Now that we ve develoed a strategy for identifying the form of a desired oint inside of a quadrant, we simly need to determine which quadrant we are in so that we can fill in the aroriate signs. Exercise : Use the unit circle on Page to record the stard radians in each quadrant. Then, identify a relationshi between the numbers in the numerator denominator. Solution: Quadrant Radians Pattern I, R.A. II 5, ONE LESS III 7, 5 ONE MORE IV 11, 7 5 ONE LESS THAN DOUBLE 5 For examle, in Exercise, we were given cos. Immediately we can identify that we are in QIV because 5 is one less than () =(i.e. the numerator is one less than double the denominator)! Therefore, the comlete oint containing cosine is given by: + 1,. 5 Consequently, cos = 1. When evaluating sin or cos, where is a stard distance/radian, we roceed as follows: 1 Identify whether the oint is on the axes or inside of a quadrant. Recall that the distances on the x axis are multiles of, on the y axis they are odd multiles of / inside the quadrants they have denominators of,, or. If inside of a quadrant, identify the R.A. for the given. Determine the quadrant by identifying the attern: R.A. (QI), ONE LESS (QII), ONE MORE (QIII), ONE LESS THAN DOUBLE (QIV) Use your knowledge of the quadrant the size of the R.A. to construct the comlete oint, beginning with the sine value, getting cosine by default. Making Math Possible 7 of 10 c Sa diyya Hendrickson

8 Worked Examles Level: x + y =1 5 Exercise 5: Evaluate sin Exercise : Evaluate cos Exercise 7: Evaluate cos 5 Making Math Possible 8 of 10 c Sa diyya Hendrickson

9 Evaluating Trig Inverses Level: x + y =1 Things to Remember 1 The range domains of inverses switch! So while sine cosine eat radians (distances travelled) return x y coordinates, their inverses eat coordinates return the radians/distances that we were required to travel. Because sine, cosine tangent are not 1 1 over their domains, their inverses are only valid on the restricted intervals given in their definitions. Thus, when evaluating, we must adhere to these restrictions. Exercise 8: Evaluate arccos Making Math Possible 9 of 10 c Sa diyya Hendrickson

10 Evaluating Trig Inverses Level: x + y =1 Exercise 9: Evaluate arcsin Making Math Possible 10 of 10 c Sa diyya Hendrickson