"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

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Trigonometry:

Lesson 1:

• If you want to solve an equation using your graphing calculator there are two methods:

  • make one side equal to zero, graph the equation and trace the x-intercepts​

  • make one side equal to y1 and the other side equal to y2 and trace the intersection points

• A general solution tells you the solution to an equation over the entire domain of real numbers.

  • look at the solution over a restricted domain (like from 0 to 360) then look for repetition over that restricted domain to find out if you can simplify by adding a general extension

    • for example, if the solution is where sin(x) = 1/2 then the solution on the restricted domain is 30 degrees and 150 degrees, these will repeat for every full revolution we make around the unit circle, so the general solution is 30 + 360n and 150 + 360 n, where n is an integer.

• A non-permissible value (NPV) is a value that is not allowed to be part of the solution set. These are values that will make the denominator of the question equal to zero (and you're not allowed to divide by zero)

  • Find the NPV's on a restricted domain, than use the same method as described above to add a general extension to that the restriction is true across the domain of all real numbers. ​

Lesson 2:

• When you have to solve a trigonometric equation algebraically, always check to see if you can factor it (either take out a common factor,  or treat it like a quadratic and factor into binomials)

  • each factor is equal to zero, use the factors to find the "roots" ​or solutions to the equation

  • Don't forget to check your equation for NPV's and to extend the solution across the domain of real numbers (find the general solution)

Lesson 3:

• The three primary trigonometric ratios have reciprocal identities (csc(x), sec(x), cot(x))

• If a question asks you to verify, it is asking you to substitute some value into the equation to show that it is balanced

• If a question asks you to prove then you are trying to balance the equations without cross multiplying

  • choose one side of the equation that is the most simple, try to get the other side to look the same as the simple side​

  • put your equation into the trigonometric building blocks if possible (that means make all of the trig ratios into cosine and sine using identities)

  • remember that AN EXAMPLE IS NOT A PROOF

  • always check for NPV's based on the ORIGINAL QUESTION and remember to look at where the denominators are worth zero and where the denominators are undefined (if your denominator could be expressed as a fraction)

Lesson 4:

• The unit circle and trigonometry is all about right-angled triangles, so you can use the Pythagorean Theorem to solve almost anything!

• Know the three basic Pythagorean identities on page 9 of your notes, and how to derive the two other forms of each identity

• Whenever you have a squared trigonometric ratio and you are adding or subtracting the number one, you might be able to substitute a Pythagorean identity.

Lesson 5:

• When you find yourself in a situation where you are not dealing with a "nice" angle (one that matched one of our three special triangles) you will likely either have to use the sum and difference identities or the double angle formulas (see lesson 6)

• Use your knowledge of fractions to create any fraction from one of the "nice" angles through addition (preferred) or subtraction

• Use the sum and difference identities to substitute your unknown angle situation with something that you are familiar with

• DON'T FORGET: sometimes you will be given a separated set of angles and will have to use them to find a total sum or difference --> equal signs in identities work in both directions!

Lesson 6:

• Similar to lesson 5. If you have an angle that is unfamiliar, and the angle measure is even, you might have a double angle situation!

• An exact value is a fraction, when a question asks for the exact value, the answer better be in fraction-form

• Pay attention to domain restrictions, if you have enough information then you can draw a representative right-angle triangle to get more information (about the angle or a missing side)

• DON'T FORGET: sometimes you will be given something that looks like the right-hand side of the double-angle identities and you will have to replace them wit cos(2x) or sin(2x) or tan(2x) --> equal signs in identities work in both directions!

Lesson 7:

• You can use identities to express equations in terms of a single trigonometric ratio (this will help with factoring and finding solutions)

• Once you have an equation written using only one type of trig ratio, then you can either factor out a common term or factor a quadratic-style equation into two binomials.

• Make your factors equal to zero and solve for the solution to the equation

• Remember to check for non-permissible values, and to give your solution in general form if a domain is not specified.