Transcription of TRIGONOMETRY NOTES By STEVEN SY Copyright 2008
1 TRIGONOMETRY NOTESBySTEVEN SYCopyright Facts .. Formulas ..91 Review of .. and Range of Functions .. Transformations .. of Functions; Even and Odd .. of Functions .. of Combined Functions .. of Composition: From Output Formulas .. of Composition: From Graphs .. One-to-One Functions .. Inverse Functions .. Inverse Functions II: Reflections .. Domain and Range of the Inverse Function .. Capital Functions .. 782 Rational Reciprocal Function .. Functions and Asymptotes .. Rational Functions .. 963 Elementary and Revolutions .. Wrapping Function.
2 Wrapping Function At Multiples of and .. Wrapping Function At Multiples of .. Wrapping Function At Multiples of and .. Trigonometric Functions: Definitions .. and Range of the Trigonometric Functions .. Functions: Periodicity .. Functions: Even/Odd Behavior .. Elementary Trigonometric Relationships .. 1654 Graphing Trigonometric of Sine and Cosine .. Sinusoids .. Phenomena .. of Other Trigonometric Functions .. General Tangent and Cotangent .. General Secant and Cosecant .. Trigonometric Functions .. Harmonic Motion and Frequency .. 2105 Trigonometric Trigonometric Relationships.
3 Trigonometric Identities .. and Difference Formulas I .. and Difference Formulas II .. Formulas .. Formulas .. Relationships and Formulas .. More Trigonometric Identities .. to Sum Formulas .. Sum to Product Formulas .. Verifying Even More Trigonometric Identities .. 2746 Advanced Trigonometric Trigonometric Functions .. Trigonometric Problems I .. Trigonometric Problems II .. Trigonometric Functions .. Trigonometric Problems .. Inverse Trigonometric Identities .. Trigonometric Identities .. Trigonometric Equations I .. Trigonometric Equations II .. Harmonic Combination.
4 3557 Triangle Angles .. Triangle TRIGONOMETRY .. and Angles .. Triangle Formulas and Derivations .. Triangle Types .. Oblique Triangles .. of a Triangle .. 415 Selected Answers to the Basic Facts1. DO NOT BLINDLY APPLY powers and roots across expressions that have or As in comment 1, is something that can NOT be simplified!!3. As in comment 1, can not be done without square formula applies: . Notice the term. This means when yousquare you will have a term that looks like twice the product of the terms in parentheses. You getthis from In particular, . Do NOT forget the middle term. Note that you canget this quickly by multiplying and and Factoring FormulasA.
5 FormulasPerfect Square Factoring: Difference of Squares: Difference and Sum of Cubes: B. Comments1. There is no sum of squares formula, no formula for (over the real numbers).2. With and in the same equation, you get one equation when you take the top signs,and you get another when you take the bottom you get and .3. The easy way to remember the Difference and Sum of Cubes Formula is to rememberthat the first factor looks like you just remove the cubes. Then the second factor lookslike you square the first factor, except rather than doubling the middle term, you take thenegative of the middle ExamplesExample 1:Factor Solution Now use the Perfect Square Formula (with minus):Ans Example 2:Factor Solution Now use the Difference of Squares Formula:Ans 10 Example 3:Factor Solution Now use the Sum of Cubes Formula:Ans Example 4:Factor Solution Now use the Difference of Cubes Formula:Ans 11 Exercises1.
6 Expand .2. Expand .3. Factor .4. Factor .5. Factor .6. Factor .7. Factor .121314 Chapter 1 Review of FunctionsA. Definition of a FunctionEvery valid input, , producesexactly oneoutput, ; no more, no lessB. Explicit vs. Implicit Functions:function whose defining equation is solved for . Functions:function whose defining equation isnotsolved for .15C. ExamplesDetermine if the following equations define functions of ; if so, state whether they areexplicit or 1: SolutionPlug in some -values, see how many -values you get: For each , we only get one y, so this a function of .AnsThis is an explicit function of .Example 2: Solution .We have two s.
7 So this is not a function of .AnsThis is not function of .16 Example 3: Solution ..For each , we only get one y, so this a function of .AnsThis is an implicit function of .Note:If we have a graph, we may determine if we have a function of by using theVertical Line Test(if any vertical line hits the graph more than once, it is not afunctionof ). For example: nota function of 17D. Notation and Comments 1. The function operator is written incursiveto distinguish it from a : input function operator;representsthe function; eats to spit back output same as y; output of the functionNote: isnotthe function, represents it. is a -value. For instance, if 3 is aninput, is the output ( -value).
8 3. In terms of , is theformula for the Evaluation ExamplesConsider Example 1:Find SolutionWe want the output when . Use formula for the output, , and plug in .Now , so .Ans21 Example 2:Find and simplifySolutionWe want the output when the input is . Plug into formula foroutput where you see :Now , so .Then simplify: Ans 19 Example 3:Simplify thedifference quotient SolutionNow so .Then Ans , if 20 Exercises1. Determine if the following equations define functions of ; if so, state whether they are explicitor b. c. d. e. f. g. 2. Let . Find and simplify:a. b. c. d. Note: This is different than part f. 213. Let . Find and simplify:a. b. c.
9 D. 4. Find the difference quotient and simplify for .5. Find the difference quotient and simplify for .6. Find and simplify for . Domain and Range of FunctionsA. Domain all valid inputsB. Range all outputsC. Finding DomainWe throw away all problem particular, we don t allow division by zero or complex things to : Throw away values making the denominator Roots: Set inside , and solve : Set inside , and solve Domain Finding ExamplesExample 1:Given , find .SolutionNothing in checklist, so domain is all real Example 2:Given , find . :Throw away . Root:Set Throw away Ans 24E. Finding RangeThis is more By plugging in different -values, try to see what -values you get back.
10 Whatis the smallest -value? What is the largest -value? Are any -values missed?Heuristic:Expressions that are raised to even powers or even roots of ex-pressions have smallest -value equal to Graph it, and read off the -values from the See if you can apply HSRV transformations to a known basegraph (reviewedlater in Section )4. For a quadratic function, find the vertex. Depending on whether the parabolaopens up or down, the -value of the vertex will give you the minimum or themaximum value of the range, Odd degree polynomials have range .There are other methods, such as the Back Door method, which will not be Range Finding ExamplesExample 1:Given , find.
