Math12 lesson 6

2,065 views

Published on

Published in: Education, Technology
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
2,065
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
80
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • Week 5
  • 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œƒ.<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 />2𝑠𝑒𝑐2πœƒ=11βˆ’sinπœƒ+11+sinπœƒ<br />1+cosπœƒcosπœƒ=secπœƒ+1<br />cosπœƒsecπœƒ+tanπœƒ=1βˆ’sinπœƒ<br />1tanπœƒ+cotπœƒ=sinπœƒΒ cosπœƒ<br />2π‘π‘œπ‘ 2πœƒβˆ’1=π‘π‘œπ‘ 4πœƒβˆ’π‘ π‘–π‘›4πœƒ<br />tanπ‘₯+cotπ‘₯=cscπ‘₯secπ‘₯<br />Β <br />
    9. 9. sinπœƒβˆ’tanπœƒ2=π‘‘π‘Žπ‘›2πœƒcosπœƒβˆ’12<br />cosπœƒ1+sinπœƒ+cosπœƒ1βˆ’sinπœƒ=2cosπœƒ<br />sinπœƒ+cosπœƒ2tanπœƒ=tanπœƒ+2 𝑠𝑖𝑛2πœƒ<br />1+sinπœƒ+cosπœƒ1βˆ’sinπœƒβˆ’cosπœƒ=βˆ’2sinπœƒcosπœƒ<br />π‘‘π‘Žπ‘›2πœƒ1βˆ’π‘π‘œπ‘ 2πœƒ+π‘ π‘–π‘›πœƒπ‘ π‘’π‘2πœƒβˆ’1=π‘π‘œπ‘ πœƒπ‘ π‘’π‘3πœƒ+π‘π‘œπ‘‘πœƒ<br />π‘π‘œπ‘ 2π‘₯+1+sinπ‘₯π‘π‘œπ‘ 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œ‹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œ‹βˆ’π‘₯sinπ‘₯+sinπœ‹βˆ’π‘₯cosπ‘₯<br /> c) tan49Β°+tan23Β°1βˆ’tan49Β°tan23Β°<br />Find the exact value of a) sinπ›Όβˆ’π›½ and b) π‘‘π‘Žπ‘›π›Ό+𝛽 if sin𝛼=βˆ’35Β Β Β and sin𝛽=15; the terminal side of 𝛼 lies in Q3 and the terminal side of 𝛽 lies in Q1.<br />Verify: sinπ‘₯βˆ’πœ‹2=cosπ‘₯+πœ‹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‘₯=βˆ’25 and πœ‹<π‘₯<3πœ‹2, find sin2π‘₯.<br />Simplify each expression and evaluate the resulting expression exactly, if possible.<br /> a) 2tan15Β°1βˆ’π‘‘π‘Žπ‘›215Β° b) π‘π‘œπ‘ 2π‘₯+2βˆ’π‘ π‘–π‘›2π‘₯+2<br />Verify each identity.<br /> a) sinπ‘₯+cosπ‘₯2=1+sin2π‘₯<br /> b) sin3π‘₯Β =sinπ‘₯4π‘π‘œπ‘ 2π‘₯βˆ’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œ‹8 c) sin75Β°<br />2. If cscπ‘₯=βˆ’3 and cosπ‘₯>0, find cosπ‘₯2.<br />If cotπ‘₯=βˆ’245Β Β Β Β Β andΒ Β Β πœ‹2<π‘₯<πœ‹,Β Β Β findΒ Β sinπ‘₯2.<br />Verify the following:<br /> a) sinβˆ’π‘₯=βˆ’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) 4cosβˆ’π‘₯cos2π‘₯ d) sinβˆ’πœ‹4π‘₯cosβˆ’πœ‹2π‘₯<br />Write each expressions as a product of sines and/or cosines:<br /> a) cos2π‘₯βˆ’cos4π‘₯Β  c) sin0.4π‘₯+sin0.6π‘₯<br /> b) sinπ‘₯2βˆ’sin5π‘₯2 d) cosβˆ’πœ‹4π‘₯+cosπœ‹6π‘₯Β <br />Β <br />
    19. 19. Examples<br />Simplify the following trigonometric expressions:<br /> a) cos3π‘₯βˆ’cosπ‘₯sin3π‘₯+sinπ‘₯ b) cos5π‘₯+cos2π‘₯sin5π‘₯βˆ’sin2π‘₯<br />Verify the following:<br /> a) 𝑠𝑖𝑛 𝐴+sin𝐡cos𝐴+cos𝐡=π‘‘π‘Žπ‘›π΄+𝐡2<br /> b) sinπ΄βˆ’sinπ΅π‘π‘œπ‘ π΄+π‘π‘œπ‘ π΅=π‘‘π‘Žπ‘›π΄βˆ’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 />

    Γ—