Sum-Difference+Identities

toc SUM-DIFFERENCE IDENTITIES: media type="custom" key="5952979"media type="custom" key="5889857" ​​ =__Sum, Differ____ence, & Double Angle Identities__ =

♦
 * Angle Sum and Difference Theorem:** The following identities are true for all values for which they are defined:

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

cos(x + y)=cos(x)cos(y) -- sin(x)sin(y)

sin(x - y)=sin(x)cos(y) -- cos(x)sin(y)

sin(x + y)=sin(x)cos(y) + cos(x)sin(y)

gsdfdgdfg tan(x)+tan(y) tan(x+y)= -- as asdf 1-tan(x)tan(y)

sdfggsdfg tan(x)-tan(y) tan (x-y)= -- sssssss 1+tan(x)tan(y)

Although the equations look long, it is really just simply a matter of plugging in the numbers.

Don't get too worked up about the x's and y's, you could even substitute them for A and B if you'd like. As long as x sine and x cosine are from the same number and the y sine and y cosine are from the same, you should be good to go.

Here is an example of proving sum and difference identities.

cos(x-y) + cos(x+y) = 2cos(x)cos(y) cos(x)cos(y) + sin(x)sin(y) + cos(x)cos(y) - sin(x)sin(y) = 2cos(x)cos(y) 2cos(x)cos(y) = 2cos(x)cos(y)

--->> Here is a "cheat sheet" for when you start using Tangent from the Unit Circle in the formulas state above. Don't forget!!! To find tangent, the formula is tan=y/x, (meaning tangent = y over x).


 * Degrees || Radians || Tangent Measure ||
 * 30° || π /6 || (√3)/3 ||
 * 45° || π /4 || 1 ||
 * 60° || π /3 || (√3) ||
 * 90° || π /2 || Undefined ||
 * 120° || 2π /3 || (-√3) ||
 * 135° || 3π /4 || -1 ||
 * 150° || 5π /6 || (-√3)/3 ||
 * 180° || π || 0 ||
 * 210° || 7π /6 || (√3)/3 ||
 * 225° || 5π /4 || 1 ||
 * 240° || 4π /3 || (√3) ||
 * 270° || 3π /2 || Undefined ||
 * 300° || 5π /3 || (-√3) ||
 * 315° || 7<span style="font-family: 'Times New Roman',Times,serif;">π /4 || -1 ||
 * 330° || 11<span style="font-family: 'Times New Roman',Times,serif;">π /6 || (-√3)/3 ||
 * 360° || 2<span style="font-family: 'Times New Roman',Times,serif;">π || 0 ||

=media type="custom" key="5948037" align="center"=

= Unit Circle UNIT CIRCLE =

__Proof Practice__
//Proof:// Let //P// be the point with coordinates. Measured counterclockwise from point //P//, let //Q// be the point whose arc length is //A//, let //R// be the point whose arc length is, and let S be the point whose arc length is. Then are the coordinates of point //Q//,  are the coordinates of point //R//, and  are the coordinates of point //S//. Note that the lengths of segments //PR// and //QS// are equal. Using the distance formula, we get: Through the use of the [|symmetric] and Trigonometric Identities, this simplifies to become the angle sum formula for the cosine.

The formulas can also be derived using triangles. Although we refer to the following derivation as a proof, in fact the values of angles //A// and //B// allowed by the derivation are quite limited, and a more general proof is actually required media type="youtube" key="PA1IwrsDv1E" height="389" width="496" align="center"

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Angle Difference Formula
These formula allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the cos(15°). Unlike the cos(60°) which can be expressed as ½, the cos(15°) cannot simply be represented as a rational expression. However, the angle difference formula allows you to represent the exact value of this function. = **NOW YOU TRY IT! Try these sample problems. Just highlight the text to see the answer!** = cos15=?
 * sin(A − B) = sinAcosB − cosAsinB
 * cos(A − B) = cosAcosB + sinAsinB


 * cos15=cos(60-45)
 * We know that cos(x-y)=cosxcosy+sinxsiny, so
 * cos(60-45)=cos60cos45+sin60sin45
 * cos(60-45)=(1/2)(√2/2)+(√3/2)(√2/2)
 * cos(60-45)=((√2)+√6)/2

=Angle Sum Formula= These formulas allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the sin(105°). Unlike the sin(30) which can be expressed as ½, the sin(105) cannot simply be represented as a rational expression. However, the angle sum formula allows you to represent the exact value of this function
 * sin(A+B) = sinAcosB + cosAsinB
 * cos(A+B) = cosAcosB − sinAsinB

<span style="color: #59ff00; font-family: 'Comic Sans MS',cursive; font-size: 150%;">So to wrap things up: media type="youtube" key="v4JrGnWgJLg" height="485" width="792" align="left"