Quick Instructions
Add Verifying & Simplifying Using Trigonometric IDs content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.

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

Red - cosecant Blue - sine

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.

There is symmetry at the origin. 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.

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.

Quick InstructionsAdd Verifying & Simplifying Using Trigonometric IDs content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.

Verifying and Simplifying Using Trigonometric IDs (Unit 7A) Learning TargetsThe Trigonometric IdentitiesRed - secant

Pink - cosine

Red - cosecant

Blue - sine

The Reciprocal IdentitiesThe Quotient Identities(also called Negative Angle or Opposite Angle IDs)The Even/Odd IdentitiesThese identities help determine if the trigonometric function is an odd function or even function.There is symmetry at the origin.There is symmetry along the y-axis.Example of odd functionNotice 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.

Example of an even functionNotice 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.Pythagorean IdentitiesRemember as "

1tanin asecond"Remember as "

1 cottage in thecascades"The Co-function IdentitiesThe Sum/Difference IdentitiesHere'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 IdentitiesWant to see how double/ angle identities were derived?

http://www.themathpage.com/atrig/double-proof.htm

Example:Answer:ORORExample:Answer:Example:Answer:Simplifying TipsExample1) 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 TipsExample1) 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: