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Verifying and Simplifying Using Trigonometric IDs (Unit 7A) Learning Targets
  • Develop and apply trigonometric identities (reciprocal, quotient, negative angles, Pythagorean, cofunctions, sum/difference, and double angle)
  • Apply trigonometric identities to simplify expressions and verify equations


‍‍‍The Trigonometric Identities‍‍‍
Red - secant


Pink - cosine

otherg7.gif


Red - cosecant
Blue - sine


otherg2.gif



‍‍‍otherg91.gif‍‍‍



otherg94.gif


The Reciprocal Identities







The Quotient Identities





‍‍‍The Even/Odd Identities‍‍‍ ‍‍‍(also called Negative Angle or Opposite Angle IDs)‍‍‍
These identities help determine if the trigonometric function is an odd function or even function.


external image o13.gif
There is symmetry at the origin.

external image e38.gif
There is symmetry along the y-axis.










Example of odd function
Notice that when a sin(x) graph is reflected across the x-axis and then the y-axis, the resulting graph is identical to a regular sin(x) graph. When a sin(x) graph is reflected across the y-axis, the resulting graph appears to be a sin(x) graph reflected across the x-axis.
Screen_shot_2012-05-30_at_3.12.22_PM.png








Example of an even function
Notice that when a cos(x) graph is reflected across the x-axis and then the y-axis, the resulting graph appears to be a cos(x) graph reflected over the x-axis. Notice that when a cos(x) graph is reflected across the y-axis, the resulting graph is identical to a regular cos(x) graph.
Screen_shot_2012-05-25_at_8.52.26_PM.png









‍‍‍Pythagorean Identities

external image picture-graph-of-pythagorean-identity-formula.gif





‍‍Remember as "1 tan in a second"‍‍‍


Remember as "1 cottage in the cascades"‍‍‍


The Co-function Identities
Remember, π/2 - θ is always the opposite
Remember, π/2 - θ is always the opposite



‍‍‍The Sum/Difference Identities
‍‍‍
Here's the rap we heard in class to help us remember these identities:
http://www.youtube.com/watch?v=0SzufD96p58

Want to see how the identities are derived?
http://www.maa.org/pubs/mm_supplements/smiley/trigproofs.html

To help you remember:
http://www.youtube.com/watch?v=aFrRTnsDxms



Example:


Answer:




Example:


Answer:





Example:


Answer:




Example:


Answer:




Example:


Answer:




Example:


Answer:



‍‍Double/Angle Identities‍‍
Want to see how double/ angle identities were derived?
http://www.themathpage.com/atrig/double-proof.htm



Example:


Answer:



OR

OR


Example:


Answer:




Example:


Answer:


Simplifying Tips
  • simplify based on Even-Odd Identities
  • rewrite based on QIDs and RIDs
  • look for/recognize other trig IDs
  • check to make sure your final answer can't be simplified any more

Example
simplify1.JPG
1) split fraction


2) cancel terms

3) PIDs

4) final answer





Note: There are multiple ways to solve this problem! (for example, instead of splitting fractions use PIDs to make numerator cot^2(x))


Verifying Tips
  • Wh‍‍en verifying, try to work on only one side
  • OR work both sides separately
  • Basically, verifying is just like simplifying. Except there are two sides of an equation that must end up the same
  • IMPORTANT: When verifying, you CANNOT operate over the equal sign. EVER.

Example‍‍
verify1.JPG
1) QIDs (numerator) and PIDs (denominator)


2) RIDS (denominator)


3) reduce terms

4) use double angle IDs


5) final answer!



Video Links:

Test your knowledge! Quiz yourself!
http://www.proprofs.com/quiz-school/story.php?title=trigonometric-identity-quiz

Derivations video:
http://www.youtube.com/watch?v=OLzXqIqZZz0

Double angle problems:
http://www.youtube.com/watch?v=7Eo-fuy0f7g

Sum and Difference problems:
http://www.youtube.com/watch?v=ZhvvkCa_60w

Some simplifying examples:
http://www.youtube.com/watch?v=SZu_EVV4jjY
http://www.youtube.com/watch?v=FS6iQX7jY-s
http://www.youtube.com/watch?v=fkrLkvQSlmk




Original authors of this page (06/02/12): Sakthi S., Jenny Z., Rebecca K., Sanika B., Lara B., Ryan S.
Editors and secondary contributors: