How to solve trigonometric equations by factoring?

Solving trigonometric equations by factoring isn't hard at all, it is very very similar to solving algebraic equations by factoring. This is how we start if we were giving this problem:

sinxcosx-3cosx=0 

We first factor out cosine, and we get cosx(sinx-3). Then, we set both cosx to 0 and sinx-3 to zero and than we solve. We should get sinx equals 3 and cosx equals 0. We than use arcsin and we get that cosx is equal to 90. We find all the other solutions by finding its reference angle, so that means cosx is equal to 90 and 270. Now, sinx=3 isnt possible. The unit circle only goes up to 1 because the radius is 1, therefore the only solutions are 90 and 270. 

And, viola! Thats how you solve trigonometric equations by factoring.

Try this:

sinxcosx=-cosx

How do we solve linear trigonometric equations?

Linear Trigonometric equations is a basic algebra problem such as:

3x+5=17

Just how we solve the algebra problem above by isolating the x, we do the same for trigonometric equations. However, the only difference of these equations is the memorization of the value of angles. 

For example: 

sin(x)+2=3 for 0° < x < 360°

We solve this equation just like an algebra equation. We subtract 2 from both sides and we should get sin(x)=1. This is where remembering the value of certain angles come in handy. If you do not remember the value of these angles, you can use your calculator to find out what is the value of sin(x)=1. You use arcsin in order to find out the value. This will appear like this on your calculator: sin-¹.You find the arcsin of 1 which is 90°. Than, you must find the reference angles which equal 1 as well. However, in this case the question above asks angles between 0 and 360. Therefore, the only angle that equals sin(x)=1 is 90°.

Viola! Thats how you solve Trigonometric Equations! 

Try this: 

                             2cos(2x)+1=0

Ever wondered how to convert Radians from Degrees and vice versa?

First off, what exactly is a radian?
Well, a radian is a unit of an angle which is equal to the central angle of a circle whose arc is is equal to the length of the radius.

This is exactly what I mean,

However, there is way to convert radians to degrees it is simple! In a full circle there are 2 Pi radians, therefore in half a circle (180 degrees) there are Pi radians. So in the radian measurement Pi radians is 180 degrees.

So that leaves us to this formula on converting Degrees to Radians:

And to convert Radians into degrees this shall be the formula: 

And Viola! Thats all you have to do, plug in whatever the angular measure and you are done!

Now try this:

Convert 15 Degrees: 

Convert 5Pi/6 Radians: 

Why is the name Pythagorean Identity is appropriate?

We all know that the Pythagorean theorem is written or drawn like this:

a² + b² = c²

^{The Pythagorean Identity used in Trigonometry is very similar to the Pythagorean Theorem. As we learned before, the Pythagorean Identity is:}

 Sin²x+Cos²x=1

Now lets use the same triangle with numbers: 

The Pythagorean Theorem will look like: (3)(3)+(4)(4)=(5)(5)

or 9+16=25

Now using the Pythagorean Identity: 

We have to plug in for Sin²x and Cos²x. So, Sin of x is 3/5 and Cosine of x is 4/5. However, both Sin and Cosine is raised to the second power so we square Sin and Cosine:

(3/5)(3/5)+(4/5)(4/5)

(9/25)+(16/25)=25/25,

and....

25/25=1!

Tada !!

The Pythagorean Identity is an appropriate name for it because when you plug the numbers in the Pythagorean Theorem just how it is formulated (remember Sin and Cosine were raised to the second power, don't forget to raise it to the second power!) you will result in 1, ALL THE TIME! 

If you forget the Pythagorean Identity, remember the Pythagorean theorem and you will succeed in trigonometry! 

How to solve rational equations?

Rational Equations: Rational Equations are two polynomial functions written as a ratio. For Example:

Rational equations are just like proportions, which make this quite easy to solve. All you have to do is cross multiply. In the equation above all you have to do is give the fractions a common denominator by multiplying the denominator with the numerator and denominator. After you find its common denominator you cross out the denominator and you are left with the numerator which is x²+6x=8x=24. Now you combine all like terms and you are left with a quadratic equation. Than you solve the quadratic equation and get x=6,-4.

Viola! Simple as that!

Now try this problem on your own:

Sources:

• https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcRwefBwxYVBy5zgfpLI96fhYWCofGQjqQFECXTYEK5kP1B5Sjac
• http://www.purplemath.com/modules/rational/solve04.gif

How do we multiply and divide rational expressions?

Multiplying and Dividing rational expressions are really easy. All you need to remember is to multiply straight across then simplify, simple as that! Like this:

Here you factor the expressions out first than you multiplied straight across. 4x+8 over  x²-25 times x-5 over 5x+10. You take factor each expression and you get 4(x+2) over (x+5)(x-5) times x-5 over 5(x+2). Than you cancel out what the expressions have in common like x+2 and x-5. Multiply straight across and viola! the answer is 4 over 5(x+5).

Dividing rational expressions includes a different step. In order to divide rational expressions you must KCF, which stands for Keep Change Flip. You must keep the first expression change the division sign to a multiplication sign than flip the second expression. Like this:

Here you keep the 2x²y over 3yz and change the division sign to a multiplication sign and flip the expression to 16xd²   over 9y²z. Than you cross out what is in common to simplify the expression and multiply straight across, and there is your answer!




Try This:



Sources:

• http://virtualnerd.com/images/Tutorial/Alg1_8_1_4-diagram_thumb-lg.png
• http://www.virtualnerd.com/images/Tutorial/Alg1_08_01_0019-diagram_thumb-lg.png
• http://hotmath.com/help/solutions/genericalg1/7/2/RationalExpressionsandRationalEquations/genericalg1_7_2_RationalExpressionsandRationalEquations_25_605/image062.gif

How do we simplify expressions with rational exponents?

Rational exponents are fractions, for example like this:

The numerator is the number of times the base is to be multiplied. The denominator is the root being taken.
You'll know your expression is simplified when:

1. The exponent isn't negative: You must flip the exponent and put it in the denominator and make it positive.
2. When the index are really simplified: if you can simplify the fraction exponent, do it! Its much easier!

Lets try a Problem:

Here 27 is raised to the 2/3rd power. The denominator is the root, so 27 to the cube root is 3. Than you square 3 and your final answer is 9. Viola! Simple as that.

Now Try this problem on your own:
What is 25/9 to the 1/2 Power?






Sources:

• http://www.solving-math-problems.com/image-files/math_rad_frac_exp.png
• https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQ5GY6vvbHS0glig6_EY2DjtvWd45kOcuOvBBgbKuY9xhD06OR9
• http://www.mathops.com/free/a1rr010.php