TRIGONOMETRIC IDENTITIES. An identity is an equation that is true for all defined values of a variable. We are going to use the identities that we have already established and establish others to "prove" or verify other identities. Let's summarize the basic identities we have. - PowerPoint PPT Presentation
Text of TRIGONOMETRIC IDENTITIES
An identity is an equation that is true for all defined values of a variable.We are going to use the identities that we have already established and establish others to "prove" or verify other identities. Let's summarize the basic identities we have.
RECIPROCAL IDENTITIESQUOTIENT IDENTITIES
Lets look at the Fundamental Identity derived in Section 1.6Now to find the two more identities from this famous and oft used one.Divide all terms by cos2xcos2xcos2xcos2xWhat trig function is this squared?1What trig function is this squared?Divide all terms by sin2xsin2xsin2xsin2xWhat trig function is this squared?1What trig function is this squared?These three are sometimes called the Pythagorean Identities since the derivation of the fundamental theorem used the Pythagorean Theorem
All of the identities we learned are found on the back page of your book.You'll need to have these memorized or be able to derive them for this course.QUOTIENT IDENTITIESPYTHAGOREAN IDENTITIESRECIPROCAL IDENTITIES
One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Lets see an example of this:substitute using each identitysimplify
Another way to use identities is to write one function in terms of another function. Lets see an example of this:This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.
We'd get csc by taking reciprocal of sinNow use the fundamental trig identitySub in the value of sine that you knowSolve this for cos When we square root, we need but determine that wed need the negative since we have an angle in Quad II where cosine values are negative.square root both sidesA third way to use identities is to find function values. Lets see an example of this:
We need to get tangent using fundamental identities.Simplify by inverting and multiplyingFinally you can find cotangent by taking the reciprocal of this answer.You can easily find sec by taking reciprocal of cos.This can be rationalizedThis can be rationalized
Now lets look at the unit circle to compare trig functions of positive vs. negative angles.Remember a negative angle means to go clockwise
Recall from College Algebra that if we put a negative in the function and get the original back it is an even function.
Recall from College Algebra that if we put a negative in the function and get the negative of the function back it is an odd function.
If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also.EVEN-ODD PROPERTIESsin(- x ) = - sin x (odd) csc(- x ) = - csc x (odd)cos(- x) = cos x (even) sec(- x ) = sec x (even)tan(- x) = - tan x (odd) cot(- x ) = - cot x (odd)