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Trigonometric Equations Using Factoring

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Proofs of Trigonometric Functions

FOIL: FOIL is a memory device that describes the process for multiplying two binomials, meaning multiplying the First two terms, the Outer two terms, the Inner two terms, and then the Last two terms, and then summing the four products.

Trigonometric Identity: trigonometric identity is an expression which relates one trig function on the left side of an equals sign to another trig function on the right side of the equals sign.

For some refreshers on trig identities click here.

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Option One: Often one of the steps for proving identities is to change each term into their sine and cosine equivalents.

Option Two: Use the Trigonometric Pythagorean Theorem and other Fundamental Identities.

Option Three: When working with identities where there are fractions- combine using algebraic techniques for adding expressions with unlike denominators.

Option Four: If possible, factor trigonometric expressions. For example, \frac{2 + 2 \cos \theta}{\sin \theta(1+ \cos \theta)} = 2 \csc \thetacan be factored to \frac{2 (1+ \cos \theta)}{\sin \theta (1 + \cos \theta)} = 2 \csc \theta and in this situation, the factors cancel each other.

TIP: Always look for things to factor out, this will often lead to the problem to be more simpler and will often reveal the answer. Also, always try to make a T chart when trying to prove the identities, it will make the work more visible and organized.

\begin{array}{c|c } \frac{1 + 1 + 2 \cos \theta}{\sin \theta(1+ \cos \theta)} & 2 \csc \theta  \\\frac{2 + 2 \cos \theta}{\sin \theta(1+ \cos \theta)} & 2 \csc \theta \\\frac{2 (1+ \cos \theta)}{\sin \theta (1 + \cos \theta)}& 2 \csc \theta \\\frac{2}{\sin \theta}& \frac{2}{\sin \theta} \end{array}



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