# algebra ii m2 topic a lesson 6 teacher

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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 - -

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

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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