Chapter 9: Trigonometric Identities and Equations Goal: Learn Reciprocal Functions and their applications inside exercises using identities Goal: Recognize substitution of Pythagorean identities Goal: Verify trigonometric identities analytically and support solutions graphically. Goal: Further develop algebra skills

9.1 Fundamental Identities Reciprocal Identities Quotient Identities Pythagorean Identities Negative-Number Identities Note It will be necessary to recognize alternative forms of the identities above, such as sin²  = 1 – cos²  and cos²  = 1 – sin² .

9.1 Rewriting an Expression in Terms of Sine and Cosine Example Write tan  + cot  in terms of sin  and cos . Leaving our final result as one fraction Solution

9.1 Verifying Identities Learn the fundamental identities. Try to rewrite the more complicated side of the equation so that it is identical to the simpler side. It is often helpful to express all functions in terms of sine and cosine and then simplify the result. Usually, any factoring or indicated algebraic operations should be performed. For example, As you select substitutions, keep in mind the side you are not changing, because it represents your goal. *occassionally it may make sense to change both sides, these are more complex* If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin² x, which could be replaced with cos² x. **useful trick when you recognize/determine that all terms are already in terms of sine and cosine and you’re not sure where to go.**

Additional Tips Always keep an eye out for cos2(x) or sin2(x), sec2(x) and tan2(x). This is usually an indication of a Pythagorean identity. If you change everything to sin and cos it is very important that you attempt to create one fraction on whichever side you are working on. Do not try to memorize all of your identities but learn the process for developing them. At first you may want to develop a list of identities so you can access them quickly There is more than 1 way to skin a cat.

There are usually two types of problems that you see There are usually two types of problems that you see. One will say simplify and there will usually be rules such as simplify down to one trig function or simplify down to some congruent statement. The other type of problem is referred to as verifying or proving an identity. You will basically do exactly what you do in the first style of problem but one side of the “=“ will be the rule for you to simplify the other side down to

Simplify to one trig function

Simplify

Simplify

Simplify so that sin(x) is the only trig function leftover

Simplify so that only one trig function is leftover

9.1 Verifying an Identity (Working with One Side) Example Verify that the following equation is an identity. cot x + 1 = csc x(cos x + sin x) Analytic Solution **which side looks more complicated?**

9.1 Verifying an Identity (Working with One Side) Graphing Calculator Support We can SUPPORT our identity with a graph. If you graph the side that you are working on at any juncture it should always be the exact same graph as the side that you have not changed…Remember identities are equations that are always true regardless of the independent variable chosen. Check with a TI-84 or Geogebra. (either one you must understand your syntax and order of operations) *To verify an identity, we must provide an analytic argument. A graph can only support, not prove, an identity.

9.1 Verifying an Identity Example Verify that the following equation is an identity. Solution

Practice Problems These first ones we will work together, then independently

The next two slides incorporate the process of working on both sides of the equation. At times you may not recognize that the work you have done has lead you anywhere productive. If this is the case it may seem useful to work on the other side of the equation in hopes that you are able to make the other side look like what you have transformed the first side to look like. At any point that you are able to show both sides identical, you have successfully proven the identity.

9.1 Verifying an Identity (Working with Both Sides) –save for second or third day Example Verify that the following equation is an identity. Solution

9.1 Verifying an Identity (Working with Both Sides) Now work on the right side of the original equation. We have shown that

Revisiting Initial Goals Goal: Learn Reciprocal Functions and their applications inside exercises using identities Goal: Recognize and apply substitution of Pythagorean identities Goal: Verify trigonometric identities analytically and support solutions graphically. Goal: Further develop algebra skills