Fundamental Identities

Key Questions

  • How do you use the fundamental trigonometric identities to determine the simplified form of the expression?

    "The fundamental trigonometric identities" are the basic identities:

    •The reciprocal identities
    •The pythagorean identities
    •The quotient identities

    They are all shown in the following image:

    ![academics.utep.edu)

    When it comes down to simplifying with these identities, we must use combinations of these identities to reduce a much more complex expression to its simplest form.

    Here are a few examples I have prepared:

    a) Simplify: tanx/cscx xx secx

    Apply the quotient identity tantheta = sintheta/costheta and the reciprocal identities csctheta = 1/sintheta and sectheta = 1/costheta.

    =(sinx/cosx)/(1/sinx) xx 1/cosx

    =sinx/cosx xx sinx/1 xx 1/cosx

    =sin^2x/cos^2x

    Reapplying the quotient identity, in reverse form:

    =tan^2x

    b) Simplify: (cscbeta - sin beta)/cscbeta

    Apply the reciprocal identity cscbeta = 1/sinbeta:

    =(1/sinbeta - sin beta)/(1/sinbeta)

    Put the denominator on a common denominator:

    =(1/sinbeta - sin^2beta/sinbeta)/(1/sinbeta)

    Rearrange the pythagorean identity cos^2theta + sin^2theta = 1, solving for cos^2theta:

    cos^2theta = 1 - sin^2theta

    =(cos^2beta/sinbeta)/(1/sinbeta)

    =cos^2beta/sinbeta xx sin beta/1

    =cos^2beta

    c) Simplify: sinx/cosx + cosx/(1 + sinx):

    Once again, put on a common denominator:

    =(sinx(1 + sinx))/(cosx(1 + sinx)) + (cosx(cosx))/(cosx(1 + sinx))

    Multiply out:

    =(sinx + sin^2x + cos^2x)/(cosx(1 + sinx))

    Applying the pythagorean identity cos^2x + sin^2x = 1:

    =(sinx + 1)/(cosx(1 + sinx))

    Cancelling out the sinx + 1 since it appears both in the numerator and in the denominator.

    =cancel(sinx + 1)/(cosx(cancel(sinx + 1))

    =1/cosx

    Applying the reciprocal identity 1/costheta = sectheta

    =secx

    Finally, on a last note, I know that here in Canada, British Columbia more specifically, these identities are given on a formula sheet, but I don't know what it's like elsewhere. In any event, many students, me included, memorize these identities because they're that important to mathematics. I would highly recommend memorization.

    Practice exercises:

    Simplify the following expressions:

    a) cosalpha + tan alphasinalpha

    b) cscx/sinx - cotx/tanx

    c) sin^4theta - cos^4theta

    d) (tan beta + cot beta)/csc^2beta

    Hopefully this helps, and good luck!

    Noah G · 3 · Aug 8 2016
  • What are even and odd functions?

    Even & Odd Functions

    A function f(x) is said to be {("even if "f(-x)=f(x)),("odd if "f(-x)=-f(x)):}

    Note that the graph of an even function is symmetric about the y-axis, and the graph of an odd function is symmetric about the origin.

    Examples

    f(x)=x^4+3x^2+5 is an even function since

    f(-x)=(-x)^4+(-x)^2+5=x^4+3x^2+5=f(x)

    g(x)=x^5-x^3+2x is an odd function since

    g(-x)=(-x)^5-(-x)^3+2(-x)=-x^5+x^3-2x=-f(x)

    I hope that this was helpful.

    Wataru · 4 · Nov 12 2014
  • How do you use the fundamental identities to prove other identities?

    Divide the fundamental identity sin^2x + cos^2x = 1 by sin^2x or cos^2x to derive the other two:

    sin^2x/sin^2x + cos^2x/sin^2x = 1/sin^2x

    1 + cot^2x = csc^2x

    sin^2x/cos^2x + cos^2x/cos^2x = 1/cos^2x

    tan^2x + 1 = sec^2x

    Kevin B. · 1 · Feb 22 2015

Questions

Trigonometric Identities and Equations