7.1+Content+-+Verifying+&+Simplifying+Using+Trigonometric+IDs

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.
 * Quick Instructions**


 * 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

Red - secant math y=cos(x) math
 * __‍‍‍The Trigonometric Identities ‍‍‍ __**

Pink - cosine math y=sec(x) math

Red - cosecant Blue - sine math y=csc(x) math math y=sin(x) math

math y=tan(x) math ‍‍‍ ‍‍‍

math y=cot(x) math

math \frac{1}{sin(x)} = csc(x)\ math
 * __The Reciprocal Identities__**

math \frac{1}{cos(x)} = sec(x) math

math \frac{1}{tan(x)} = cot(x) math

math \frac{sin(x)}{cos(x)} = tan(x)\ math
 * __The Quotient Identities__**

math \frac{cos(x)}{sin(x)} = cot(x)\ math

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

//There is symmetry at the origin.//

//There is symmetry along the y-axis.//

math \ -sin(-x) = sin(x)\ math

math \sin(-x) = - sin(x)\ math

math \ -csc(-x) = csc(x)\ math

math \csc(-x) = - csc(x)\ math

//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.// ‍ ‍ math \ -cos(-x) = - cos(x)\ math
 * //Example of odd function//**

math \cos(-x) = cos(x)\ math

math \ -sec(-x) = - sec(x)\ math

math \sec(-x) = sec(x)\ math

//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.// math \ -tan(-x) = tan(x)\ math
 * //Example of an even function//**

math \tan(-x) = - tan(x)\ math

math \ -cot(-x) = cot(x)\ math

math \cot(-x) = - cot(x)\ math


 * __‍‍‍Pythagorean Identities__**



math \cos^{2}(x) + sin^{2}(x) = 1\ math

math \ 1 + tan^{2}(x) = sec^{2}(x)\ math ‍‍Remember as "**1** **tan** in a **sec**ond" ‍‍‍

math \ 1 + cot^{2}(x) = csc^{2}(x)\ math Remember as "**1 cot**tage in the **c**a**sc**ades**"** ‍‍‍


 * __The Co-function Identities__**

Here's the rap we heard in class to help us remember these identities: []
 * __‍‍‍The Sum/Difference Identities__**
 * __ ‍‍‍__**

Want to see how the identities are derived? []

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

math \ sin(A-B) = sin(A)cos(B) - cos(A)sin(B)\ math

math \ sin(90-75) = sin(90)cos(75) - cos(90)sin(75)\ math
 * Example:**

math \sin(15)\ math
 * Answer:**

math \ sin(A+B) = sin(A)cos(B)+cos(A)sin(B)\ math

math \sin(120+90) = sin(120)cos(90) - cos(120)sin(90)\ math
 * Example:**

math \sin(210) = \frac{-1}{2}\ math
 * Answer:**

math \ cos(A-B) = cos(A)cos(B)+sin(A)sin(B)\ math

math \ cos(150-130) = cos(150)cos(130)+sin(150)sin(130)\ math
 * Example:**

math \cos(20)\ math
 * Answer:**

math \ cos(A+B) = cos(A)cos(B)-sin(A)sin(B)\ math

math \cos(30+45) = cos(30)cos(45)-sin(30)sin(45)\ math
 * Example:**

math \ cos(75) = \frac{\sqrt{6}-\sqrt{2}}{4}\ math
 * Answer:**

math \tan(A-B) = \frac{tan(A)- tan(B)}{1 + tan(A)*tan(B)}\ math

math \tan(150-115) = \frac{tan(150)- tan(115)}{1 + tan(150)*tan(115)}\ math
 * Example:**

math \tan(45) = 1\ math
 * Answer:**

math \tan(A+B) = \frac{tan(A) + tan(B)}{1 - tan(A)*tan(B)}\ math

math \tan(45+135) = \frac{tan(45)+tan(115)}{1 - tan(45)*tan(115)}\ math
 * Example:**

math \tan(180) = 0\ math
 * Answer:**

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

math \ sin(2x)=2sin(x)cos(x)\ math

math \ 4sin(\frac{\Pi }{6})*cos(\frac{\Pi }{6})\ math
 * Example:**

math \ 2sin(\frac{\Pi }{3})=2(\frac{\sqrt{3}}{2})=\sqrt{3}\ math
 * Answer:**

math \ cos(2x)=cos^{2}(x)-sin^{2}(x) \ math math \ 2cos^{2}(x)-1\ math math \ 1-2sin^{2}(x)\ math
 * OR**
 * OR**

math \ 2cos(2x)=cos^{2}(\frac{\Pi }{4})-1\ math
 * Example:**

math \ cos(2*\frac{\Pi }{4})=cos(\frac{\Pi }{2})=0\ math
 * Answer:**

math \ tan(2x)=\frac{2tan(x)}{1-tan^{2}(x)}\ math

math \ 2tan(\frac{(\frac{\Pi }{2})}{1-tan^{2}(\frac{\Pi }{2})})\ math
 * Example:**

math \ tan(2^{\frac{\Pi }{2}})=tan(\Pi )=0\ math
 * 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

1) split fraction
 * Example**

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.

1) QIDs (numerator) and PIDs (denominator)
 * Example** ‍‍

2) RIDS (denominator)

3) reduce terms

4) use double angle IDs

5) final answer!

__Video Links:__

Test your knowledge! Quiz yourself! []

Derivations video: []

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: [] [] []


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