Trig Functions and Co-functions

         There is nothing more confusing then trig, it is a well known fact. Sine, Cosine, and Tangent seem pretty easy to understand, but other functions stem from these. Now introducing Co-secant, Secant, and Cotangent. These are known as co - functions because they correspond to one of the main trig functions. Sine and Co-secant are co functions, Cosine and Secant are co - functions. And as you may have guessed Cotangent and Tangent are co - functions. Now co - functions are just 1 over the main function. for instance Cotangent is 1/tan or since tan is sin/cos it is cos/tan. Now to simplify these kinds of problems you must put everything in terms of sin and cos. For example:

csc-tan-cos

Putting everything in terms of sin and cos:

1/sin-sin/cos-cos

Now cancel out:

     Both the sin's in the numerator and denominator cancel out and you place cos over one at the end and then both cos's cancel out laving you with:

1/1

Final answer: Because one divided by one is still one your final answer is one. Simple.

      

        Now sometimes these problems are squared. These are still solved in the exact same way but is your final answer was to be cos it would become cos^2. Just make sure everything you cancel out makes sense. for example if your final answer is 1/cos that can be simplified to just sec. 

             

    Happy Trig-ing!

How do we solve linear trigometric equations?

         Don't we all wish we can go back to the days where all we had to solve was 2x+7=14? Good and bad news. Good being we get to go back to these kinds of problems

Now the bad news is that it involves trig. Don't be scared yet, its the same basics as solving a regular linear equation. These problems tend to look like: Sin(∅)+2=3. 

    The first step to solving this is to get sin alone. So we subtract two from both sides. Once two is subtracted you are left with: sin(∅)=1. This is where our handy dandy unit circle comes into play. We look to see at what angle sin equals 1 which only happens at 90° this means our answer is 90°. 

      Like everything in math, these problems can get a lot more complex and can have multiple answers. For example if after you've simplified your answer is cos(∅)=√2/2 then you are looking for every where that cosine = √2/2. In this case that would be both at 45° and 315°.

     These problems aren't that hard to figure out, with a little practice you will know where every single angle is, no problem! 

How do we slove reference angle problems?/How to convert between radians & degrees.

     When solving for reference angles its always best to use your unit circle. Your reference angle is between 0 and 90 degrees and you're trying to find the quickest way back to the x-axis. 

      Solving a reference angle problem can be very tricky the best way is to subtract 360° from the angle given. For example if you are given 345° and are asked to find the reference angle you do 345 - 360 and get -15. You take away the negative sign and your reference angle is 15°. Your reference angle is never negative and never more than 90°. 

      Sometimes you maybe asked to solve a reference angle in radians, which looks like 18π/3. The easiest way to solve these are to convert to degrees, solve, and then convert your answer back into radians. To convert to degree you simply divide and multiply times 180. Then to convert back you DIVIDE by 180, simplify, and add π. There's nothing to it!

Why is the Title Pythagorean Identify Appropriate?

Pythagorean Identity is an appropriate title for many reasons. Sine and Cosine are the root of all identies and all others come from these two. When defining 'identity' it can be defined as, an equation that is true no matter what values are chosen. The original Pythagorean Identity is sin^2 θ + cos^2 θ = 1. In the unit circle the hypotenuse is always 1 and the x and y. values are cosine and sine. If you look at the Pythagorean Therom (a^2 + b^2 = c^2) you can see that compared to the Pythagorean Identity it is basically the same thing. Because C is the hypotenuse in the Identity it becomes 1. And cosine and sine are just a and b. Because these two are so related that is why the Pythagorean Identity is an appropriate title. No matter what values you chose it will always be true.

How do we simplify complex fractions?

            Complex fractions can be very intimidating, but once you get used to them there a breeze. Lets take a look at one:

1

_

x

__

1

_

x+1

      Now the first step is to just rewrite the equation so it looks like a simple division problem. So it's(1/x)/(1/x+1). Now with all division problems we use Keep. Flip. Change. We keep 1 over x, flip the second half so it is x+1 over 1, and change the sign to division sign to multiplication.

      The last thing you do is cancel out what you can in this case that would be the ones. You are left with a final answer of:

x+1

____

x

       It really is that simple and straight forward. You're all done!

How do we solve rational equations?

          The single thing that complicates solving rational equations is getting that common denominator. Much like adding and subtracting rational expressions the denominators must be the same. Let's take a look at an example:
                                         
        Our common denominator is going to be 5x(x+2) so you have to multiply each part of the equation so that they each have the same denominator. Multiply the first section by 5x, the second by 5(x+2), and the third by x+2. After all of that your new equation will look like this:
                            
             At this point, the denominators are the same. So do they really matter? Not really (other than for saying what values x can't be). So cross out the denominators and you have an equation that looks like: 15x–(5x + 10) =x+2. The rest is cake we are trying to get x alone so you solve like a regular equation. First distribute. Then The rest looks like:
                                      15x – (5x + 10) = x +2
                                          10x – 10 = x + 2
                                                9x = 12 
                                            x = ¹²/₉ = ⁴/₃
                 Since x = ⁴/₃ won't cause any division-by-zero problems in the fractions in the original equation, then this solution is valid. And that's all there is to it!

How do we multiply and divide rational expressions?

          When working with regular fractions, multiplying and dividing is much easier than adding and subtracting. The sames rules apply for simplifying rational expressions. Lets take a look at an example:      

                                                     

      Our first step is to completely factor the numerator and denominator. The first numerator factors to (x-2)(x-6) and the first denominator is a perfect square so that factors into (x+4)(x-4). The second numerator factors into 4(x+4) and the second denominator factors into (x-2)(x-2). This leaves you with (x-2)(x-6)/(x+4)(x-4) times 4(x+4)/(x-2)(x-2). 

       The next step is to cancel out all the terms that are the same, like this: 

                                     

      You can only cancel out terms that are exactly the same like (x-2) on top and ONE of the (x-2)s on the bottom. One you have canceled all things the same on the top and bottom you combine what you have left as your final answer. In the numerator you were left with 4 and (x-6) and in the denominator you were left with (x-4) and (x-2). You put all of hat together like the last section shown above and you have your final answer. Much easier than it looks. 

      Just remember to can not cancel TERMS only FACTORS. Meaning if we go back to our original equation you can not just cancel x²   in the numerator and denominator. You must factor first.

      When dividing rational expressions you do the same exact thing you do when multiplying. The key to dividing is to remember Keep. Flip. Change. Keep the first set the same, flip the second set, and change the operation to multiplication. Then you factor and cancel out like you normally would.