How to Verify & Simplify Trig Expressions

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Verifying and simplifying trig expressions will make certain expressions easier to calculate. Verify and simplify trig expressions with help from an experienced mathematics professional in this free video clip.

Part of the Video Series: Trigonometry, Graphs, & Other Math Tips
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Hello, my name is Walter Unglaub. And this is how to verify and simplify trig expressions. So, a trigonometric expression, such as sine of theta can be written in terms of exponential using Euler's formula. As E to the I theta minus E to the minus I theta, all over two times I. Where I in this expression is the imaginary unit, which is equal to the square root of negative one. And cosine of theta can be written in a similar fashion as E to the I theta plus E to the minus I theta, all over two. It's important to also understand the relation between these two very commonly used trig expressions and tangent of theta. Tangent of theta is simply the ratio of sine of theta divided cosine of theta. So, if you you understand these expressions, then you can verify more complicated looking expressions. And for the next part of this, I'm going to utilize a simple example to demonstrate how one can simplify expressions, understanding the relationship between tangent, sine and cosine. So, let's say I have a function of theta which is equal to cosine of theta cubed. I'm not going to write theta here for the sake of simplicity. Times tangent of theta plus cosine of theta. And I divide by sine squared of theta times cotangent squared theta. Well, cotangent is simply one over tangent. So, I can rewrite this in terms of just sines and cosines as cosine cubed times sine divided cosine, plus cosine in the numerator. And in the denominator, I'll have sine squared times cosine squared, divided sine squared. So, immediately I can see that I can cancel out these sine squared terms. And in the numerator, I can cancel out this cosine with one of these cosines. So, I have a two in the exponent for this cosine. So, this is equal to cosine squared times sine, plus cosine, all over cosine squared. And these are all functions of theta. So, the next step is, I can just simply this by putting in terms of two fractions, So, I have for my first fraction, cosine squared sine, divided cosine squared. The cosine squared terms cancel out. And then, for the second term, I have cosine from the numerator, divided by the common denominator, cosine squared. And this cosine cancels out with one of the cosines at the bottom. So, this is equal to sine plus one over cosine. And one over cosine is equal to secant. So, I can write this in a slightly simpler form, as sine of theta plus secant of theta. So, understanding the relationships between sine, cosine, tangent, can allow one to simplify complicated looking expressions into simpler trig expressions. My name is Walter Unglaub, and this is how to verify and simplify trig expressions.

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