THREE

 This chapter is all about Trigonometric Identities. 

 __Section 3.1__ Reciprocal Identities:

Quotient Identities:

Pythagorean Identities:

Odd and Even Identities:



[|Identities Video]

 Practice: [|Practice sites:]  [|More Practice]

Section 3.2 // An expression involving one or more trigonometric functions could be written in many different equivalent forms. We must often decide whether two expressions are equivalent and we do this through an identity. This section concentrates on techniques for verifying an identity. // Practice [|Verifying Identities]

Section 3.3 //In every identity discussed in Sections 3.1 and 3.2, the trigonometric functions were functions of only a single variable, such as x. In this secion we etablish identities for the cosine of a sum or difference of two variables. They do not follow from the known identities but rather from the geometry of the unit circle.//

__**Identities**__  Cosine of a Sum cos(A+B)=cosAcosB-sinAsinB  Cosine of a difference cos(A-B)=cosAcosB+sinAsinB

__**Cofuntions**__ cosx=sin(90degrees-x) sinx=cos(90degrees-x) tanx=cot(90degrees-x) cotx=tan(90degrees-x) secx=csc(90degrees-x) cscx=sec(90degrees-x)

[|Practice with cofunctions]

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Section 3.4 ** // In the last secion we learned the identities for the cosine of a sum or difference. Unfortunately they are not as simple as we might like them to be. The same situation holds for the sum and difference identities for the sine and tangent functions, which we study next. //  ** Identities **

Sine of a sum sin(A+B)=sinAcosB+cosASinB

Sine of a difference sin(A-B)=sinAcosB-cosAsinB

Examples: sin(139degrees)cos(4degrees)-cos(139degrees)sin(4degrees) sin(139degrees-4degrees) sin(135degrees) (sqrt2/2)

sin(195) sin(150+45) sin150cos45+cos150sin45 (1/2)(sqrt2/2)+(-sqrt3/2)(sqrt2/2) sqrt2/4 + ((-sqrt3 sqrt2)/4) (sqrt2-sqrt6)/4 Tangent of a sum tan(A+B)=(tanA+tanB)/1-tanAtanB

Tangent of a difference tan(A-B)=(tanA-tanB)/1+tanAtanB

EXAMPLE:  tan(pi/12) tan(pi/12)=tan(pi/3-pi/4) =[tan(pi/3)-tan(pi/4)]/[1+tan(pi/3)tan(pi/4) =[sqrt3-1]/[1+sqrt3*1] =[(sqrt3-1)(sqrt3-1)]/[(sqrt3+1)(sqrt3-1)] =[4-2sqrt3]/2 =2-sqrt3



Section 3.5 //<span style="FONT-SIZE: 110%; FONT-FAMILY: Arial, Helvetica, sans-serif"> In Sections 3.3 and 3.4 we studied identities for functions of sums and differences of angles. The double-angle and half-angle identities, which we develop next, are special cases of those identities. These special cases occur so often that they are remembered as separate identities. // ** Double-Angle Identities


 * Half-Angle Identities

**   **      or  or  or  ||
 * **Half Angle Identities** ||
 * [[image:http://www.mathwords.com/h/h_assets/half%20angle%20identities%20sin%20with%20root.gif width="161" height="46" align="left"]] or [[image:http://www.mathwords.com/h/h_assets/half%20angle%20identities%20sin%20squared.gif width="145" height="39" align="absMiddle"]]

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