1 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II : Why Call I Lesson 6 t Tangent? Student Outcomes Students define the tangent function and understand the historic reason for its name. 30° Students use special triangles to determine geometrically the values of the tangent function for , 45° , and 60° . Lesson Notes 휃 ) ° ( sin ( ) , of an acute angle , tan 휃 ratio the to is extended = ° 휃 In this lesson, the right triangle definition of the tangent cos ( 휃 ° ) ) ( cos where all real numbers tangent function defined for . ≠ ° 휃 0 The word tangent already has geometric meaning, 휃 so the historical reasons for naming this particular function are investigated . Additionally, the correlation of tangent MP.7 ( ) tan with the slope of the line that coincides with the terminal ray after rotation by 휃 휃 is noted . These three ° degrees & MP.8 different interpretations of the tangent function can be used immediately to analyze properties and compute values of the tangent fu nction. Students look for and make use of structure to develop the definitions in Exercises 7 and 8 and using what they know about the sine and cosine functions applied look for and express regularity in repeated reasoning to the tangent function in Exerc ise 3 . s This lesson depend on vocabulary from G eometry such as secant lines and tangent lines. The terms provided are used in this lesson and in subsequent lessons. for reference : (description) , UNCTION F ANGENT T tangent function The { | 180 푥 ∈ ℝ tan 푥 ≠ 90 + : 푘 for all integers 푘 } → ℝ In the . 푘 for all integers , 푘 180 + 90 ≠ 휃 be any real number such that 휃 can be defined as follows: Let Cartesian plane, rotate the initial ray by 휃 degrees about the origin. Intersect the resulting terminal ray with the 푦 휃 . unit circle to get a point 휃 is ) ° ( ( tan . The value of ) 푥 푦 , 휃 휃 푥 휃 The following trigonometric identity, ) sin ( 휃 ° 90 푘 ≠ 휃 = ) ° 휃 ( tan for all 180 + , for all integers 푘 , cos ( 휃 ° ) ) 휃 ( sin ° ° ) = or simply, 휃 ( tan , should be talked about almost immediately and used as the working definition of ) cos ( 휃 ° tangent. : A secant line to a circle is a line that intersects a circle in exactly two points. S ECANT TO A C IRCLE : A IRCLE tangent line to a circle is a line in the same plane that intersects the circle at one and only T ANGENT TO A C one point. Why Call It Tangent? Lesson 6 : 83 This work is licensed under a s Thi work is derived from Eureka Math math.org - eureka Minds. t a e Gr ©2015 ™ Minds. reat G by licensed and ShareAlike 3.0 Unported License. - NonCommercial - Creative Commons Attribution - - - This file derived from ALG II - TE 1.3.0 M2 08.2015

2 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II wor Class k Opening Exercise (4 minutes) ( ) The scription of tan xercise leads students to the de 휃 ° E pening as the quotient of 푦 and 푥 using right triangle O 휃 휃 trigonometr y. Opening Exercise ( ) 풙 be the point where the terminal ray intersects the unit circle after rotation by Let 풚 , 휽 푷 휽 휽 Scaffolding: degrees, as shown in the diagram below. for this For students not ready level of abstraction, use 휃 = 30 for this example 휃 instead of the generic value . in terms of 휽 ? a. Using triangle trigonometry, what are the values of 풙 풚 and 휽 휽 ( ) ( ) = 풚 and , ° 휽 퐬퐢퐧 퐜퐨퐬 = 풙 휽 ° 휽 휽 ) ( Using triangle trigonometry, w 풚 and 풙 in terms of b. ? ° 휽 퐭퐚퐧 hat is the value of 휽 휽 풚 휽 ( ) ° 휽 = 퐭퐚퐧 풙 휽 ) ( 휽 퐭퐚퐧 What is the value of c. ? 휽 in terms of ° ) ( ° 휽 퐬퐢퐧 ) ( = ° 휽 퐭퐚퐧 ) ( ° 휽 퐜퐨퐬 Discussion minutes) 6 ( were extended to the sine and cosine In the previous lessons, the idea of the sine and cosine ratios of a triangle the number of degrees of rotation of the initial ray in the coordinate represents , that 휃 functions of a real number, xtended to the plane. In the following discussion, similarly the idea of the tangent ratio of a n acute angle of a triangle is e ( ° ) 휃 sin ) ( ° on a subset of the real numbers. = 휃 tan tangent function ° ( cos ) 휃 MP.3 In this discussion, students should notice that the tangent ratio of an angle in a triangle does not extend to the entire real line because we need to avoid division by zero. Encourage students to find a symbolic representation for the points excluded from the domain of the tangent function; that is, the tangent function is defined for all real numbers 휃 except 180 = 휃 90 + . 푘 , for all integers 푘 Why Call It Tangent? Lesson 6 : 84 This work is licensed under a is t a e - math.org Gr Minds. ©2015 Minds. reat eureka by licensed and ™ Math Eureka from derived work s Thi G ShareAlike 3.0 Unported License. Creative Commons Attribution - NonCommercial - 1.3.0 This file derived from ALG II - 08.2015 - M2 - TE -

3 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II As the discussion pr ogresses , refer frequently to the image of the unit circle with the initial ray along the positive 푥 - axis and the terminal ray intersecting the unit circle at a point 푃 with coordinates ( 푥 , 푦 ) , the Opening in was done as 휃 휃 Exercise. ts to draw similar diagrams in their own notes as well. Encourage studen Discussion A description of the tangent function is provided below. Be prepared to answer questions based on your understanding of this function and to discuss your responses with others in your class. degrees about the origin. Let 휽 be any real number. In the Cartesian plane, rotate the nonnegative 풙 - axis by 휽 ( 퐭퐚퐧 then the value of , ퟎ , 풙 ( Intersect the resulting terminal ray with the unit circle to get a point ° ) ≠ 풙 . If ) 풚 휽 휽 휽 휽 풚 ( ) 퐬퐢퐧 ° 휽 휽 ( ) ( ) 퐜퐨퐬 for = ° 퐭퐚퐧 is . ퟎ ≠ . In terms of the sine and cosine functions, ° 휽 휽 퐜퐨퐬 ( 휽 ° ) 풙 휽 Scaffolding: To recall some of the information students have developed in the last few lessons, drawing the unit circle on the board with a reference angle and sine and cosine labeled may be helpful. A picture is included. ) ° 휃 ( sin for ( = ) ° 휃 ( tan We have defined the tangent function to be the quotient cos 휃 ° ) ≠ 0 . Why do we cos ) ° 휃 ( ) ° ( cos specify that ? 0 ≠ 휃 We cannot divide by zero, so the tangent function cannot be defined where the denominator is zero. ( ) 휃 and which segment has Looking at the unit circle in the figure, which segment has a measure equal to sin ° , a measure equal to ) ° 휃 ( cos ? ) ( ) ( 휃 sin = 푃푄 ° 휃 cos = 푂푄 and , . ° ) ( tan to be undefined. ° 휃 (Scaffolding: When that cause 휃 identify several values of , Looking at the unit circle 푥 zero?) coordinate of point - the is 푃 ( ) ( ) , 0 = ° 휃 휃 tan cos When ° then is undefined, which happens when the terminal ray is vertical so that degrees of rotation locate the terminal ray numbers of The following axis. - 푦 lies along the 푃 point , , 90 axis: - 푦 along the − 270 , 450 . 90 ( ) 휃 Describe all numbers for which cos 휃 ° = 0 . 푘 + 90 , for any integer 푘 180 Lesson 6 : Why Call It Tangent? 85 This work is licensed under a reat Eureka Math ™ and licensed by G Thi Minds. work s is derived from math.org - eureka Minds. t a e Gr ©2015 - ShareAlike 3.0 Unported License. Creative Commons Attribution - NonCommercial - - 08.2015 - 1.3.0 M2 TE - This file derived from ALG II

4 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II ? 0 ≠ ) ° 휃 ( cos with 휃 How can we describe the domain of the tangent function, other than all real numbers 푘 The domain of the tangent function is all real numbers 휃 such that 휃 ≠ 90 + 180 . 푘 for all integers , Exercise 1 ( 8 minutes) Have students work in pai rs or small groups to complete this table and answer the questions that follow. Then debrief the groups in a discussion. Exercise 1 ( ) in the table below, use the given values of 휽 For each value of 1. ° to two and 퐜퐨퐬 ( 휽 ° ) ) to approximate 퐭퐚퐧 휽 휽 ° ( 퐬퐢퐧 . decimal place s 휽 ) ( ( ) ( ) 휽 ° 퐬퐢퐧 휽 퐜퐨퐬 퐭퐚퐧 ° 휽 ° ( degrees ) − ퟎ . ퟗퟗퟗퟗퟗퟖ ퟎ ퟖퟗ − ퟓퟕퟐ . ퟗퟔ − ퟎퟎퟏퟕퟓ . ퟗ . ퟎ − − . ퟗퟗퟗퟖ ퟎ . ퟎퟏퟕퟓ − ퟓퟕ . ퟐퟗ ퟖퟗ ퟎ ퟗퟗퟔ . ퟎ − − ퟒퟑ . ퟏퟏ − ퟎퟖퟕ . ퟖퟓ ퟎ ퟗퟖ . ퟎ − ퟖퟎ − ퟓ ퟔퟕ . − ퟏퟕ . . − ퟕퟑ . ퟏ − ퟓퟎ ퟎ ퟖퟕ . ퟎ − ퟔퟎ − ퟖퟒ . ퟎ − ퟕퟕ . ퟎ ퟔퟒ . ퟎ − ퟒퟎ ퟗퟒ ퟐퟎ − ퟑퟔ . ퟎ − − . ퟎ ퟑퟒ . ퟎ . ퟎ ퟎ ퟏ ퟎퟎ ퟎ ퟎ . ퟑퟒ ퟎ . ퟗퟒ ퟐퟎ ퟎ . ퟑퟔ . ퟒퟎ ퟎ . ퟔퟒ ퟎ ퟕퟕ ퟎ . ퟖퟒ ퟏ ퟓퟎ . ퟎ ퟖퟕ . ퟎ ퟔퟎ ퟕퟑ . ퟎ ퟓ ퟔퟕ . ퟏퟕ . ퟗퟖ . ퟎ ퟖퟎ ퟖퟓ ퟎ ퟗퟗퟔ . . ퟎ ퟎퟖퟕ ퟏퟏ . ퟒퟑ ퟎퟏퟕퟓ ퟐퟗ . ퟓퟕ ퟖퟗ . ퟎ ퟗퟗퟗퟖ . ퟎ ퟖퟗ . ퟓퟕퟐ ퟎퟎퟏퟕퟓ . ퟎ ퟗퟗퟗퟗퟗퟖ . ퟎ ퟗ ퟗퟔ . a. As 휽 → − ퟗퟎ ° and 휽 − ퟗퟎ ° , what value does 퐬퐢퐧 ( 휽 ° ) approach? > ퟏ − ) As → − ퟗퟎ ° and 휽 휽 > − ퟗퟎ approach? b. ° 휽 ( 퐜퐨퐬 , what value does ° ퟎ ) ( 퐬퐢퐧 ° 휽 ) ( 퐭퐚퐧 , how would you describe the value of ° ퟗퟎ − > 휽 and ° ퟗퟎ − → 휽 ? As c. ° 휽 = ( ) ° 퐜퐨퐬 휽 − → ) ° 휽 ( 퐭퐚퐧 ∞ Why Call It Tangent? : Lesson 6 86 This work is licensed under a work is derived from Eureka Math ™ and licensed by Thi G reat Minds. ©2015 Gr e a t Minds. eureka - math.org s ShareAlike 3.0 Unported License. - - Creative Commons Attribution NonCommercial TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

5 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II ( ) , what value does < 휽 and ° ퟗퟎ → 휽 As d. 퐬퐢퐧 휽 ° ° approach? ퟗퟎ ퟏ 휽 e. As → approach? ) ° 휽 ( 퐜퐨퐬 , what value does ° ퟗퟎ < 휽 and ° ퟗퟎ ퟎ ( ) 퐬퐢퐧 휽 ° ° ° and 휽 < ퟗퟎ → , how would you describe the behavior of 퐭퐚퐧 ( 휽 ° ) = ퟗퟎ f. As 휽 ? ) ( 휽 ° 퐜퐨퐬 ) ( ∞ ° 퐭퐚퐧 → 휽 g. How can we describe the range of the tangent function? which is The range of the tangent function is ( − ∞ , all real numbers. the set of ∞ , ) minutes) Example 1 (2 a concrete example of Now that the domain and range of the tangent function has been established , go through is used 30 = 휃 ; here 휃 computing the value of the tangent function at a specific value of , . With students use either 푦 ) sin ( 휃 ° 휃 ) ( ( ) , whichever seems more appropriate tan = 휃 ° ° = 휃 as a working definition for the tangent function tan or 휃 ° ) cos ( 푥 휃 for a given task . Example 1 Suppose that point 푷 is the point on the unit circle obtained by rotating the initial ray through ퟑퟎ ° 퐭퐚퐧 ( ퟑퟎ . Find ) . ° Scaffolding: For struggling students, provide a review of the side lengths of and - 60° - 90° 30° triangles. 90° - 45° - 45° ? 푂푃푄 △ of 푂푄 What is the length of the horizontal leg √ 3 = Geometry , we have 푂푄 By remembering the special triangles from . 2 ? 푂푃푄 △ What is the length 푄푃 of the vertical leg of Geometry Either by the Pythagorean theorem, or by remembering the special triangles from , we have 1 푄푃 = . 2 Lesson 6 : Why Call It Tangent? 87 This work is licensed under a Thi s - eureka Minds. t a e Gr ©2015 Minds. reat G work by licensed and ™ Math Eureka from derived is math.org - NonCommercial - Creative Commons Attribution ShareAlike 3.0 Unported License. TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

6 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II What are the coordinates of point ? 푃 √ 1 3 ) , ( 2 2 ( ) ( ) What are and sin 30° 30° ? cos √ 3 1 ( ( ) ) = cos 30° . , and sin = 30° 2 2 ) ( What is ? 30° tan √ 1 3 ( ) ( ) denominator = . W ith no radic als in the tan , this is tan 30° 30° = 3 √ 3 Exercise 2 – ( 8 minutes) : Why Do We Call I t Tangent? 6 In this set of exercises, begin to answer the question posed in the lesson’s title : Why Call I t Tangent? students Ask sin ( 휃 ° ) ( ) students if they can see any reason to name the function . the tangent function = ° 휃 It is unlikely that they 푓 cos ( ° ) 휃 will have a reasonable answer. E xercises 2 – 6 2. 푶 Let 푷 be the point on the unit circle with center that is 휽 the intersection of the terminal ray after rotation by 푸 degrees as shown in the diagram. Let be the foot of the line perpendicular 퓵 the line axis, and let - 풙 to the 푷 from ( be icular to the 풙 - axis at 푺 be the line perpend ퟏ , ퟎ ) . Let 푹 퓵 . point where the secant line the 퐎퐏 intersects the line ̅ ̅ ̅ ̅ 푹푺 Let 풎 be the length of . ) ( Show that . ° 풎 = a. 퐭퐚퐧 휽 ̅ ̅ ̅ ̅ has length 풎 , adjacent to and side 푶푺 Segment 푹푺 ∠ 푹푶푺 has length ퟏ , so we use tangent: 풎 퐭퐚퐧 휽 ( ) = ° = 풎 ퟏ ( ) 퐭퐚퐧 Thus, 휽 풎 = . ° Scaffolding: It may help students who mathematicians named the make a conjecture why b. U sing a segment in the figure, struggle to see the ) ( 휽 ° 퐬퐢퐧 ( ) = 휽 풇 function the tangent function. ° ) ( ° 휽 퐜퐨퐬 diagram drawn both with MP.3 and without the half chord - because the value of the function is the tangent Mathematicians named the function 푃푄 drawn. length of the segment 푹푺 that is tangent to the circle. For Exercise 2, part (b), students may need to be 푹푶푺 or 푷푶푸 △ , Why can you use either triangle △ c. , to calculate the length 풎 ? reminded that the side These triangles are similar by t share a similarity (both are right triangles tha AA △ lengths of 푃푂푄 are common acute angle) ; hence, their sides are proportional. ( ) ( ) 휃 sin . cos and ° ° 휃 you Imagine that d. are the mathematician who gets to name the function . ( ow cool would that be?) Based H ion instead? upon what you know about the equations of lines, what might you have named the funct I would have called it the slope function instead because the slope of the secant line . 풎 푶푹 is also Why Call It Tangent? Lesson 6 : 88 This work is licensed under a by licensed and ™ Math Eureka from derived is work s Thi math.org - eureka Minds. t a e Gr ©2015 Minds. reat G NonCommercial ShareAlike 3.0 Unported License. - Creative Commons Attribution - - TE - 1.3.0 - 08.2015 This file derived from ALG II - M2

7 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II ( ) 3. to illustrate what happens to the value of 퐭퐚퐧 Draw four pictures similar to the diagram 휽 ° in Exercise 2 as the rotation of the terminal ray increases towards ퟗퟎ ° . How does your diagram relate to the secant line through the work done in Exercise 1? MP.7 & MP.8 to positive infinity ퟗퟎ to As the rotation increases , the value of 풎 increases degrees . Since 풎 = 퐭퐚퐧 ( 휽 ° ) , the value ( ) 퐭퐚퐧 . 휽 ° . 1 This is what was observed numerically in Exercise of to positive infinity also increases and the tangent line When the terminal ray is vertical, what is the relationship between the secant line 푶푹 4. 푹푺 ? ) ( in this instance. What is the value of 풎 퐭퐚퐧 Explain why you cann ퟗퟎ ° ot determine the measure of ? The secant line and tangent line are parallel when the terminal ray is vertical. The value of 풎 cannot be determined because parallel lines do not intersect and triangle trigonometry does not apply. , formed is no triangle Therefore, . . The tangent function is undefined at 휽 = ퟗퟎ secant line and the 5. When the terminal ray is horizontal, what axis? Explain - 풙 is the relationship between th 푶푹 e ( ) what happens to the value of 풎 ퟎ ° 퐭퐚퐧 ? in this instance. What is the value of 푶푹 he secant line When t is horizontal, then it 푹 is zero since the points 풎 The value of axis. - 풙 coincides with the ) ( . ퟎ = ° ퟎ and 퐭퐚퐧 푺 , ퟎ = 풎 , are the same. Then and thus the , what is ° ퟒퟓ by about the origin 6. counterclockwise When the terminal ray is rotated relationship between the ̅ ̅ ̅ ̅ 풎 퐭퐚퐧 ( ퟒퟓ ? What is the value of 푶푺 length of the and ° value of ? ) ̅ ̅ ̅ ̅ ) ( ퟏ . Thus, 퐭퐚퐧 is a radius of the unit circle Since . 푶푺 = 풎 In this case, ퟒퟓ ° , 풎 = ퟏ . = 푶푺 Lesson 6 : Why Call It Tangent? 89 This work is licensed under a licensed by G reat Minds. ©2015 Gr e a t Minds. eureka - math.org ™ Thi s work is and derived from Eureka Math ShareAlike 3.0 Unported License. - NonCommercial - Creative Commons Attribution This file derived from ALG II 08.2015 - - TE - M2 - 1.3.0

8 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II While ing debrief this set of exercises, make sure to emphasize the following points: , the length of the tangent segment formed by intersecting the terminal ray degrees 90 to 0 For rotations from ( ) with the line tangent to the unit circle at ) 0 is equal to tan , 휃 ° 1 . ( when 90° = 휃 . This fact can now be related to fact that the terminal ray is undefined The tangent function 1 ( the line tangent to the unit circle at and 90 degree rotation thus after a will be parallel ) 0 , , a tangent ; segment for this rotation does not exist. = 0° is 휃 0 because the point where the terminal ray intersects the The value of the tangent function when ( ) and the distance between a point and itself is 1 0 t , , 0 . angent line is the point (9 minutes) 8 – 7 Exercises Scaffolding: Students who are struggling to ) ( the relationship between 휃 ° discover tan In these exercises, students and the slope of remember the sine values may axis through the origin that makes an angle of 휃 degrees with the for 푥 - the secant line be encouraged to recall the The interpretation of rotations that place the terminal ray in the first and third quadrants . MP.7 0 4 1 3 2 √ √ √ √ √ 휃 as the slope of this secant line provides a geometric the tangent of explanation why the , , , , sequence 2 2 2 2 2 o the fundamental fundamental period of the tangent function is as opposed t , 180° as these are the values of sine period of 360° for the sine and cosine functions. and , 60 90 at 0 , 30 , 45 , Students should work in collaborative groups or with a partner on these exercises. Then degrees. results and provide time for students to revise what they as a whole group, debrief the ially. wrote init 8 – Exercises 7 of Rotate the initial ray about the origin the stated number of degrees. Draw a sketch and label the coordinates 7. point 푷 ray intersects the unit circle. What is the slope of the line containing this ray? terminal where the ퟒퟓ b. ° ퟔퟎ ° ퟑퟎ c. a. ° ퟏ ퟐ √ − ퟑ ퟎ √ ⁄ − ퟑ 풚 풚 √ ⁄ ퟏ ퟐ ퟐ ퟐ ퟐ = = = 풎 풎 = ퟏ = = = 풎 ퟑ √ 풙 − 풙 ퟑ ퟑ ퟏ ퟏ ퟐ √ ퟐ √ ⁄ ⁄ − ퟎ ퟐ ퟐ ퟐ Lesson 6 : Why Call It Tangent? 90 This work is licensed under a work s Thi Math G math.org - Eureka from derived is ™ and licensed by reat Minds. ©2015 Gr e a t Minds. eureka ShareAlike 3.0 Unported License. - NonCommercial Creative Commons Attribution - - 1.3.0 - 08.2015 This file derived from ALG II - M2 - TE

9 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II ( ) to 퐭퐚퐧 . How do your answers ° ퟔퟎ 퐭퐚퐧 , and ) ° ퟒퟓ ( compare , ) ° ퟑퟎ ( 퐭퐚퐧 e definition of tangent to find Use th d. your work in parts (a) – (c)? √ ퟑ That is, 퐭퐚퐧 The slopes and the ( ퟑퟎ ° ) = values of the tangent functions at each rotation were the same. , ퟑ ( ) ( ) 퐭퐚퐧 = ퟏ , and 퐭퐚퐧 ퟒퟓ ퟔퟎ ° ° = ퟑ . √ If the initial ray is rotated 휽 degrees about the origin, show that the slope of the line containing the terminal e. ) ( 퐭퐚퐧 ° 휽 . Explain your reasoning. ray is equal to ) ( ) ( ) ( ° 퐜퐨퐬 = 휽 풚 , 풙 will always intersect the unit circle at The terminal ray 퐬퐢퐧 , and will always pass ° 휽 ( ) 휽 휽 ( ) ퟎ , ퟎ . Thus, the slope of the line through the origin, containing the terminal ray will always be given by ( ) ( ) ° 휽 퐬퐢퐧 ퟎ 휽 ° 퐬퐢퐧 − ) ( 풎 = = 퐭퐚퐧 . 휽 ° = ( ( ) ) ퟎ 퐜퐨퐬 퐜퐨퐬 − 휽 ° ° 휽 Now that you have shown that the f. erminal ray, equal to the slope of the t tangent function is value of the tangent function you prefer would or slope function ? Why do you think we use tangent using the name instead of as the name of the tangent function? slope include familiarity or comfort with one of the . P ossible reasons Answers may vary for the first question ideas of trigonometry and geometry. concepts. Tangent is probably used ins tead of slope due to the historical Rotate the initial ray about the origin the stated number of 8. degrees. Draw a sketch and label the coordinates of point 푷 where the terminal ray intersects the unit circle. How does your diagram in this e xercise relate to the diagram in the ? corresponding part of Exercise 7 ° What is 퐭퐚퐧 ( 휽 for these ) 휽 ? values of a. ퟐퟏퟎ ° is th From the picture, we can see that e the secant same line produced by a rotation by ퟑퟎ ° , √ ퟑ ) ( ( ) = 퐭퐚퐧 퐭퐚퐧 ퟐퟏퟎ ° ° . = so ퟑퟎ ퟑ b. ퟐퟐퟓ ° From the picture, we can see that the secant is the same line produced by a rotation by ퟒퟓ ° , ( ) ( ) so = 퐭퐚퐧 = . ퟏ ° ퟐퟐퟓ ° 퐭퐚퐧 ퟒퟓ Why Call It Tangent? : Lesson 6 91 This work is licensed under a work is derived from Eureka Math ™ and licensed by Thi G reat Minds. ©2015 Gr e a t Minds. eureka - math.org s ShareAlike 3.0 Unported License. - - Creative Commons Attribution NonCommercial TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

10 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 M2 ALGEBRA II c. ° ퟐퟒퟎ From the picture, we can see that the secant is the , ° ퟔퟎ same line produced by a rotation by ) ( ) ( so ퟐퟒퟎ = ퟔퟎ ° 퐭퐚퐧 = 퐭퐚퐧 . ° ퟑ √ d. (c) suggest about the value of the tangent function after rotating an – the results of parts (a) What do additional ퟏퟖퟎ degrees? ퟏퟖퟎ + If the initial ray is rotated by degrees, then the terminal ray is on the same line as the terminal ray 휽 he slope of the line containing when the initial ra degrees. Since t 휽 the two terminal rays is the y is rotated by and 휽 both tangent function for value of the the same, the same. is 휽 + ퟏퟖퟎ e. What is the period of the tangent functio n ? Discuss with a classmate and write your conclusions. degrees The period is ퟏퟖퟎ since rotation by ퟏퟖퟎ rotate s a line to itself. ( ) ° Use the results of Exercise 7 e ) to explain why 퐭퐚퐧 f. ퟎ ( = ퟎ . The slope of any horizontal line is zero. ) e ( Use the results of Exercise 7 g. 퐭퐚퐧 is undefined. ) ° ퟗퟎ ( to explain why The slope of any vertical line is undefined. Closing (3 minutes) In this lesson, we saw three ways to interpret the tangent function: ) ° 휃 ( sin ) ( ) ( cos , where . 0 ≠ We have a working definition of tangent as 1. = 휃 ° 휃 tan ° 휃 ( cos ) ° ) ( tan contained in the line 휃 Using similar triangles, we found that ° 2. = 푚 , where 푚 is the length of the line segment between the point ( 1 , 0 ) and the point of intersection of the terminal ray tangent to the unit circle at and ℓ ( 1 , 0 ) ℓ . line ) ( Applying the formula for slope, we see that tan 푚 휃 ° 3. , where 푚 is the slope of the secant line that contains the = 휃 degrees . terminal ray of a rotation by ) ( ge of this new tan 휃 ° in this lesson along with the domain and ran Have students summarize these interpretations of function, as well as any other information they learned that they feel is important either as a class or with a partne r . Use this as an opportunity to check for any gaps in understanding. Why Call It Tangent? : Lesson 6 92 This work is licensed under a work is derived from Eureka Math ™ and licensed by Thi G reat Minds. ©2015 Gr e a t Minds. eureka - math.org s ShareAlike 3.0 Unported License. - - Creative Commons Attribution NonCommercial TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

11 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II Lesson Summary ) ( ° 휽 퐬퐢퐧 ) ( ( ) , where A working definition of the tangent function is 퐭퐚퐧 ≠ 휽 . 휽 ퟎ = 퐜퐨퐬 ° ° ( ) 퐜퐨퐬 휽 ° ( ) is the length of the line segment on the tangent line to the unit circle centered at ° 휽 퐭퐚퐧 The value of cle and the intersection with the secant line created by the origin from the intersection with the unit cir the ( ) . his is why we call it tangent T 풙 degrees 휽 axis rotated - ) ( 휽 axis - 풙 is the slope of the line obtained by rotating the 휽 ° 퐭퐚퐧 The value of degrees about the origin. { } | 퐟퐨퐫 ퟏퟖퟎ풌 ∈ 휽 nt function is The domain of the tange 휽 ≠ ퟗퟎ + , ℝ 퐚퐥퐥 퐢퐧퐭퐞퐠퐞퐫퐬 which is equivalent 풌 ( { ) } | . 퐜퐨퐬 ퟎ ℝ ∈ 휽 ≠ to ° 휽 The range of the tangent function is all real numbers. ° The period of the tangent function is ퟏퟖퟎ . ) ( ) ( ) ( 퐭퐚퐧 ° ퟑퟎ 퐭퐚퐧 ° ퟔퟎ ° ) ퟗퟎ ° 퐭퐚퐧 퐭퐚퐧 ) ° ퟎ ( 퐭퐚퐧 ퟒퟓ ( ퟑ √ undefined ퟎ ퟏ ퟑ √ ퟑ ( Exit Ticket 5 minutes) Lesson 6 : Why Call It Tangent? 93 This work is licensed under a Eureka Math ™ and licensed by G reat Minds. Gr ©2015 Thi s work is derived e a t Minds. eureka from - math.org Creative Commons Attribution ShareAlike 3.0 Unported License. - NonCommercial - TE This file derived from ALG II 1.3.0 - - M2 - - 08.2015

12 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II Name Date t Tangent? Lesson 6 : Why Call I Exit Ticket Draw and label a figure on the circle that illustrates the relationship of the trigonometric tangent function sin ° 휃 ) ( ) ( ( ) = ° 휃 tan . 60 1 , 0 and the geometric tangent line to a circle through the point when 휃 = ) cos 휃 ° ( Explain the relationship, labeling the figure as needed. Why Call It Tangent? Lesson 6 : 94 This work is licensed under a math.org - eureka Minds. t a e Gr ©2015 reat G by licensed and ™ Math Eureka from derived is work s Thi Minds. ShareAlike 3.0 Unported License. Creative Commons Attribution NonCommercial - - TE This file derived from ALG II - - M2 - 1.3.0 - 08.2015

13 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II Exit Ticket Sample Solutions Draw and label a figure on the circle that illustrates the relationship of the trigonometric tangent function ) ( ° 휽 퐬퐢퐧 ( ) ( ) ° and the geometric tangent line to a circle through the point ퟔퟎ = 퐭퐚퐧 . 휽 when ퟎ , = ퟏ ° 휽 ) ( 퐜퐨퐬 휽 ° Explain the relationship, labeling the figure as needed. √ ퟏ ퟑ ) ) ( ( . ° ퟔퟎ Labeling as shown, lengths are 퐜퐨퐬 = 푶푸 and , 푶푺 = = ퟏ ° ퟔퟎ = 퐬퐢퐧 = 푸푷 , ퟐ ퟐ √ ퟑ ⁄ 푸푷 푺푹 ퟐ ) ( . ퟑ thus = , 푺푹 we have , Then by similar triangles = = ° ퟔퟎ ; 퐭퐚퐧 = √ ퟏ 푶푸 ⁄ 푶푺 ퟐ Problem Set Sample Solutions ( ) , and find the value of 퐭퐚퐧 in the following right triangles. 휽 ° 1. Label the missing side lengths = ퟑퟎ a. 휽 ퟏ 1 ) ( ퟑퟎ ° 퐬퐢퐧 ퟏ ퟑ √ ퟐ ) ( = = = ° ퟑퟎ = 퐭퐚퐧 ) ( 퐜퐨퐬 ퟑퟎ ퟑ ° ퟑ ퟑ √ √ ퟐ 30° Lesson 6 : Why Call It Tangent? 95 This work is licensed under a Minds. and licensed by G reat ©2015 Gr e Thi s work is derived from a t Eureka Math ™ math.org - eureka Minds. Creative Commons Attribution - NonCommercial - ShareAlike 3.0 Unported License. - 08.2015 - This file derived from ALG II M2 - TE - 1.3.0

14 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II 휽 = ퟒퟓ b. ퟐ √ ( ) 퐬퐢퐧 ퟒퟓ ° ퟐ 1 ( ) ퟏ = = = ퟒퟓ ° 퐭퐚퐧 ) ( 퐜퐨퐬 ퟒퟓ ° ퟐ √ ퟐ 45° ퟔퟎ = 휽 c. ퟑ √ ) ( 퐬퐢퐧 ° ퟑ ퟔퟎ √ ퟐ ) ( ퟑ = = = ° ퟔퟎ = 퐭퐚퐧 √ ퟏ ) ( ퟔퟎ ퟏ 퐜퐨퐬 ° 1 ퟐ 60° Let 휽 be any real number. In the Cartesian plane, rotate the initial ray by 휽 degrees about the origin. Intersect the 2. ( ) y point 푷 with the unit circle to get 풙 , 풚 resulting terminal ra . 휽 휽 a. Complete the table by finding the slope of the line through the origin and the point 푷 . Slope 휽 , in degrees Slope 휽 , in degrees ퟎ ퟎ ퟏퟖퟎ ퟎ ퟑ ퟑ √ √ ퟐퟏퟎ ퟑퟎ ퟑ ퟑ ퟐퟐퟓ ퟒퟓ ퟏ ퟏ ퟔퟎ ퟐퟒퟎ ퟑ ퟑ √ √ Undefined Undefined ퟗퟎ ퟐퟕퟎ ퟑퟎퟎ ퟏퟐퟎ ퟑ − ퟑ − √ √ ퟏ ퟏ ퟏퟑퟓ − − ퟑퟏퟓ ퟑ ퟑ √ √ ퟏퟓퟎ ퟑퟑퟎ − − ퟑ ퟑ b. Explain how these slopes are related to the tangent function . ( ) ( ) The slope 휽 푷 . 풙 퐭퐚퐧 , 풚 is equal to ° of the line through the origin and 휽 휽 Why Call It Tangent? Lesson 6 : 96 This work is licensed under a a e Gr ©2015 Minds. reat G by licensed and ™ Math Eureka from derived Minds. math.org eureka - is work s Thi t ShareAlike 3.0 Unported License. - NonCommercial - Creative Commons Attribution 1.3.0 - - This file derived from ALG II - TE - 08.2015 M2

15 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II 퓵 centered at the origin. The line 풓 of a circle of radius Consider the following diagram 3. is tangent to the circle at ( ) . 풓 , ퟎ , so is perpendicular to the 풙 - axis 퓵 푺 풕 a. 풓 = , then state the value of If in terms of one of the trigonometric functions. ퟏ ( ) ° 휽 퐭퐚퐧 = 풕 tate the value of 풕 b. If 풓 is any positive value, then s in terms of one of the trigonometric functions. ( ) 휽 퐭퐚퐧 = 풓 ° 풕 휽 and 풓 For the given values of 풕 . , find 풓 c. 휽 = ퟑퟎ , = ퟐ ퟐ ퟑ √ ( ) = = 풕 ퟐ ° ퟑퟎ ⋅ 퐭퐚퐧 ퟑ d. 휽 = ퟒퟓ 풓 = ퟐ , ) ( ퟒퟓ 퐭퐚퐧 ⋅ ퟐ = 풕 ퟐ ퟏ ⋅ ퟐ = = ° , 휽 = ퟔퟎ e. 풓 = ퟐ ) ( ퟑ 풕 ퟐ = ퟑ = ⋅ ퟐ = ° ퟔퟎ 퐭퐚퐧 ⋅ ퟐ √ √ f. 휽 ퟒퟓ , 풓 = ퟒ = ( ) ퟒ ퟒퟓ ⋅ ퟒ = 풕 ퟏ ⋅ ퟒ = = 퐭퐚퐧 ° = 풓 = ퟑ ퟑퟎ . ퟓ g. 휽 , . ퟓ ퟑ ퟕ ퟑ ퟑ √ √ ( ) . ퟑ = 풕 ° ퟑퟎ = ⋅ ퟓ = 퐭퐚퐧 ퟑ ퟔ 풓 , = 휽 = h. ퟗ ퟎ ) ( ⋅ ퟗ = ° ퟎ ퟎ 퐭퐚퐧 ⋅ ퟗ = 풕 = ퟎ Why Call It Tangent? : Lesson 6 97 This work is licensed under a work is derived from Eureka Math ™ and licensed by Thi G reat Minds. ©2015 Gr e a t Minds. eureka - math.org s ShareAlike 3.0 Unported License. - - Creative Commons Attribution NonCommercial TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

16 CORE MATHEMATICS CURRICULUM NYS COMMON Lesson 6 M2 ALGEBRA II ퟓ = 풓 ퟗퟎ , = 휽 i. 휽 푶푹 Lines and 퓵 are distinct parallel lines when = ퟗퟎ . Thus, they will never intersect , and the line segment efined by their intersection does not exist. d j. ퟑ ퟔퟎ 풓 , = 휽 = √ ( ) ퟑ = ퟑ ° ퟔퟎ = 퐭퐚퐧 ⋅ ퟑ ⋅ ퟑ 풕 = √ √ √ k. 휽 ퟑퟎ , 풓 = ퟐ . ퟏ = ퟕ ퟏ . ퟐ ퟑ ퟐퟏ √ ( ) = 퐭퐚퐧 ⋅ = ퟑퟎ ퟏ . ퟐ = 풕 ° = ퟏퟎ ퟑ ퟏퟎ ퟑ √ √ = , 풓 휽 l. ퟑ 푨 = ) ( ) ( ⋅ , for 푨 ≠ ퟗퟎ + ퟏퟖퟎ풌 , for all integers 풌 . 퐭퐚퐧 ퟑ 푨 ° 풕 = = 퐭퐚퐧 ퟑ 푨 ° ퟑퟎ = 풃 = 풓 , 휽 m. 풃 ퟑ √ ( ) ퟑퟎ ° = 퐭퐚퐧 ⋅ 풃 = 풕 ퟑ ) ( ° 휽 퐬퐢퐧 ) ( ° 휽 = 퐭퐚퐧 Knowing that n. 풓 , for = ퟏ , find the value of in terms of one of the trigonometric functions. 풔 ( ) 휽 ° 퐜퐨퐬 ) ( ° 휽 퐭퐚퐧 풕 ( ) 휽 = . = - 퐬퐢퐧 ° triangle trigonometry, Using right 풔 풔 ퟏ 풔 ( ) ° 휽 퐭퐚퐧 ) ( 퐬퐢퐧 , , which tells us . = = ° 휽 So 풔 ( ) ( ) 휽 퐭퐚퐧 휽 ° 퐬퐢퐧 ° ) ( ( ) ( ) ( ) ° ° / 퐜퐨퐬 퐭퐚퐧 휽 ° 퐬퐢퐧 퐬퐢퐧 휽 휽 ° 휽 ퟏ ퟏ = = . = = 풔 Thus, ⋅ ( ) ) ( ) ( ( ) ) ( ° 퐜퐨퐬 퐬퐢퐧 ° 퐬퐢퐧 퐬퐢퐧 휽 ° 휽 ° 퐜퐨퐬 휽 휽 ° 휽 ퟏ So 풔 = . , ( ) 휽 퐜퐨퐬 ° ) ( ) ( ° 휽 for 퐭퐚퐧 − that Using what you know of the tangent function, show 4. 휽 ≠ ퟗퟎ + 휽 ퟏퟖퟎ 풌 , for all integers ° 퐭퐚퐧 = − . 풌 ( ) ° 휽 The tangent function could also be called the slope function due to the fact that 퐭퐚퐧 is the slope of the secant 풙 ugh the origin and intersecting the tangent line perpendicular to the line passing thro - axis. If rotation of the secant ° counterclockwise a ° 휽 line by is a clockwise rotation. 휽 – is The rotation of the secant line by , then rotation secant lines will have opposite slopes, so the tangent values will also be opposites. resulting ( ) ( ) 휽 − . 퐭퐚퐧 = ° 휽 ° 퐭퐚퐧 − Thus, Why Call It Tangent? : Lesson 6 98 This work is licensed under a work is derived from Eureka Math ™ and licensed by Thi G reat Minds. ©2015 Gr e a t Minds. eureka - math.org s ShareAlike 3.0 Unported License. - - Creative Commons Attribution NonCommercial TE 1.3.0 - 08.2015 This file derived from ALG II - M2 - -

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