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# Math12 lesson 6

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• ### Math12 lesson 6

1. 1. Lesson 6: TRIGONOMETRIC IDENTITIES<br />Math 12 <br />Plane and Spherical Trigonometry<br />
2. 2. OBJECTIVES<br />At the end of the lesson the students are expected to:<br />Review basic identities.<br />Simplify a trigonometric expression using identities.<br />Verify a trigonometric identity.<br />Apply the sum and difference identities.<br />Apply the double-angle and half-angle identities.<br />Apply the product-to-sum and sum-to-product identities.<br />
3. 3. TRIGONOMETRIC IDENTITIES<br />A trigonometric identity is an equation involving trigonometric functions that hold for all values of the argument, typically chosen to be π.<br />Β <br />
4. 4. BASIC TRIGONOMETRIC IDENTITIES<br />Reciprocal Identities<br />
5. 5. Quotient (or Ratio) Identities<br />
6. 6. Pythagorean Identities<br />Negative Arguments Identities<br />
7. 7. Guidelines for Verifying Trigonometric Identities<br />The following suggestions help guide the way to verifying trigonometric identities:<br />Start with the more complicated side of the equation.<br />Combine all sums and differences of fractions (quotients) into a single fraction (quotient).<br />Use basic trigonometric identities.<br />Use algebraic techniques to manipulate one side of the other side of the equation is achieved.<br />Sometimes it is helpful to convert all trigonometric functions into sines and cosines. <br />Note:<br />Trigonometric identities must be valid for all values of the independent variable for which the expressions in the equation are defined (domain of the equation).<br />
8. 8. Examples<br /> Verify the following identities:<br />2π ππ2π=11βsinπ+11+sinπ<br />1+cosπcosπ=secπ+1<br />cosπsecπ+tanπ=1βsinπ<br />1tanπ+cotπ=sinπΒ cosπ<br />2πππ 2πβ1=πππ 4πβπ ππ4π<br />tanπ₯+cotπ₯=cscπ₯secπ₯<br />Β <br />
9. 9. sinπβtanπ2=π‘ππ2πcosπβ12<br />cosπ1+sinπ+cosπ1βsinπ=2cosπ<br />sinπ+cosπ2tanπ=tanπ+2Β π ππ2π<br />1+sinπ+cosπ1βsinπβcosπ=β2sinπcosπ<br />π‘ππ2π1βπππ 2π+π ππππ ππ2πβ1=πππ ππ ππ3π+πππ‘π<br />πππ 2π₯+1+sinπ₯πππ 2π₯+3=1+sinπ₯2+sinπ₯<br />Β <br />
10. 10. Sum and Difference Identities <br />
11. 11. Examples<br /> 1. Find the exact value for each trigonometric expression.<br /> a) sinπ12 b) sin105Β° c) tan165Β°<br />2. Write each expression as a single trigonometric function.<br /> a) sin2π₯sin3π₯+cos2π₯cos3π₯Β Β <br /> b) cosπβπ₯sinπ₯+sinπβπ₯cosπ₯<br /> c) tan49Β°+tan23Β°1βtan49Β°tan23Β°<br />Find the exact value of a) sinπΌβπ½ and b) π‘πππΌ+π½ if sinπΌ=β35Β Β Β and sinπ½=15; the terminal side of πΌ lies in Q3 and the terminal side of π½ lies in Q1.<br />Verify: sinπ₯βπ2=cosπ₯+π2<br />Β <br />
12. 12. Double-Angle Identities<br />
13. 13. Examples<br />If cosπ₯=513 and sinπ₯<0, find a) tan2π₯ b) cos2π₯<br />If cscπ₯=β25 and π<π₯<3π2, find sin2π₯.<br />Simplify each expression and evaluate the resulting expression exactly, if possible.<br /> a) 2tan15Β°1βπ‘ππ215Β° b) πππ 2π₯+2βπ ππ2π₯+2<br />Verify each identity.<br /> a) sinπ₯+cosπ₯2=1+sin2π₯<br /> b) sin3π₯Β =sinπ₯4πππ 2π₯β1Β <br />Β <br />
14. 14. Half-Angle Identities<br />
15. 15. Examples<br />Use half-angle identities to find the exact values of the following:<br /> a) cos22.5Β° b) cot7π8 c) sin75Β°<br />2. If cscπ₯=β3 and cosπ₯>0, find cosπ₯2.<br />If cotπ₯=β245Β Β Β Β Β andΒ Β Β π2<π₯<π,Β Β Β findΒ Β sinπ₯2.<br />Verify the following:<br /> a) sinβπ₯=β2sinπ₯2cosπ₯2.<br />Β <br />
16. 16. Product-to-Sum and Sum-to-Prroduct Identities<br />
17. 17. Product-to-Sum and Sum-to-Product Identities<br />
18. 18. Examples<br />Write each expression as a sum or difference of sines and/or cosines.<br /> a) cos10π₯sin5π₯ c) sin3π₯2sin5π₯2<br /> b) 4cosβπ₯cos2π₯ d) sinβπ4π₯cosβπ2π₯<br />Write each expressions as a product of sines and/or cosines:<br /> a) cos2π₯βcos4π₯Β  c) sin0.4π₯+sin0.6π₯<br /> b) sinπ₯2βsin5π₯2 d) cosβπ4π₯+cosπ6π₯Β <br />Β <br />
19. 19. Examples<br />Simplify the following trigonometric expressions:<br /> a) cos3π₯βcosπ₯sin3π₯+sinπ₯ b) cos5π₯+cos2π₯sin5π₯βsin2π₯<br />Verify the following:<br /> a) π ππΒ π΄+sinπ΅cosπ΄+cosπ΅=π‘πππ΄+π΅2<br /> b) sinπ΄βsinπ΅πππ π΄+πππ π΅=π‘πππ΄βπ΅2<br />Β <br />
20. 20. References<br />Algebra and Trigonometry by Cynthia Young<br />Trigonometry by Jerome Hayden and Bettye Hall<br />Trigonometry by Academe/Scott, Foresman<br />Plane and Spherical Trigonometry by Paul Rider<br />