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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.

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RECIPROCAL IDENTITIES QUOTIENT IDENTITIES

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Let’s look at the Fundamental Identity derived on page 445 Now to find the two more identities from this famous and oft used one. Divide all terms by cos 2 x cos 2 x What trig function is this squared? 1 Divide all terms by sin 2 x sin 2 x What trig function is this squared? 1 These three are sometimes called the Pythagorean Identities since the derivation of the fundamental theorem used the Pythagorean Theorem

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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 IDENTITIES PYTHAGOREAN IDENTITIES RECIPROCAL IDENTITIES

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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. Let’s see an example of this: substitute using each identity simplify

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Another way to use identities is to write one function in terms of another function. Let’s 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.

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A third way to use identities is to find function values. Let’s 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.

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We'd get csc by taking reciprocal of sin Now use the fundamental trig identity Sub in the value of sine that you know Solve this for cos When we square root, we need but determine that we’d need the negative since we have an angle in Quad II where cosine values are negative. square root both sides A third way to use identities is to find function values. Let’s see an example of this:

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We need to get tangent using fundamental identities. Simplify by inverting and multiplying Finally you can find cotangent by taking the reciprocal of this answer. You can easily find sec by taking reciprocal of cos. This can be rationalized

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Now let’s look at the unit circle to compare trig functions of positive vs. negative angles. Remember a negative angle means to go clockwise

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Recall from College Algebra that if we put a negative in the function and get the original back it is an even function.

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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.

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If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also. EVEN-ODD PROPERTIES sin(- 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)

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RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES EVEN-ODD IDENTITIES

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COFUNCION IDENTITIES

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The Trigonometric Functions What about angles greater than 90°? 180°? The trigonometric functions are defined in terms of a point on a terminal side r.

The Trigonometric Functions What about angles greater than 90°? 180°? The trigonometric functions are defined in terms of a point on a terminal side r.

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