34)+Solve+Trig+Eqns+Algebraically

//__Solving Trigonometric Equations Algebraically__//

I chose to do this topic because when we learned this section in class I understood what I had to do and how to solve the equations but I still had troubles with the difficult problems. Through this project I hope to find myself getting better at solving the problems I had a hard time with. A **Trigonometric Equation** is an equation that involves trigonometric functions (example-sin,cos,tan) of unknown angles. sin(x)=1/2 x=30 degrees Trigonometric equations range between many different levels of difficulties. To solve some equations you have to know some background knowledge of trigonometry and factoring. __ Steps to Solving Trigonometric Equations __ For Example: Tan(x)=1 is already in simplest form and you can solve for X by knowing your UNIT CIRCLE or by using your calculator. X=45+360n nEZ > >  Trig Identities >
 * The first step to solving trigonometric equations is knowing if the equation is already in simplest form.
 * The second step to solving for trigonometric equations is to know the trig identities.
 * 1)  Reciprocal Identities: secx= 1/cosx, cscx= 1/sinx, cotx= 1/tanx
 * 2) Quotient Identities: tanx= sinx/cosx cotx= cosx/sinx
 * 3) Pythagorean identities:

4. Cosine Sum and Difference Identity : <span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">5. Sin Sum and Difference Identity : <span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">6. Tangent Sum and Difference Identity: <span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">7.Sin Double Angle Identity:

<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">8. Cosine Double Angle Identity:

<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">9. Tangent Double Angle Identity:



<span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: left;"> By knowing these identities and being able to use them solving difficult trigonometric equations will become more simple. <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 120%; text-align: left;">For Example: to solve //tan x= sin x// you would use the quotient and reciprocal identities and solve <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">tan x = sin x <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">tan x - sin x = 0  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">(sin x / cos x) - sin x = 0  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">sin x (( 1 / cos x) - 1) = 0  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">sin x = 0   **<span style="font-family: 'Comic Sans MS',cursive; line-height: 23px;">x = 0, 180 **  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">(1/ cos x) -1 = 0  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">sec x = 1  <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: small; line-height: 23px; text-align: left; text-decoration: line-through;">x= 0, 360 <span style="font-family: 'Comic Sans MS',cursive; font-size: medium; line-height: 23px;">Another example showing how to use identities is **<span style="font-family: 'Comic Sans MS',cursive; font-size: medium; line-height: 23px;">tan x + 1= sec x, ** <span style="font-family: 'Comic Sans MS',cursive; line-height: 23px;">to solve this problem you would use the Pythagorean identity <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 120%; text-align: left;">tan x + 1 = sec x <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">(tan x + 1 = sec x) ^2  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">tan x ^2 + 1 = sec x ^2  <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">sec x^2 = sec x ^2 <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">Both these examples show how trig identities are useful in solving trig equations.
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">The third step to solving trigonometric equations is to learn how to factor. Factoring helps put the equations into a form where you can easily solve for X. For instance, in the first example given //tan x = sin x// you would not have been able to solve for X if you did not factor the sin x out into //sin x ( 1/ cos x - 1)//. By putting the equation into two different parts you are able to set each side to zero and solve for x.

<span style="font-family: 'Comic Sans MS',cursive; line-height: 23px;">I had a difficult time learning how to factor but I find if I use the //sneaky square// trick factoring is a lot easier. If you need help on factoring then go to <span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">[] for a step by step explanation!

<span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;"> An example for solving by factoring is

<span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;"> 2sinx^2 - 5 sin x +2 = 0 I used a sneaky square to factor this in to (2 sinx -1)(sinx - 2) then set each set of parenthesis equal to zero 2 sinx - 1 = 0 sinx = 1/2 x= 30, 150 + 360 n nEZ sin x - 2 = 0 sin x = 2 x never = 2

<span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;"> When you first get a problem to solve follow these steps:


 * <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">Look for trig identities that can simplify the equation
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">If you can factor then factor!
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; line-height: 23px; text-align: left;">Once the equation is in its simplest form of the equation solve for x by using your calculator or unit circle.

<span style="display: block; font-family: 'Comic Sans MS',cursive; text-align: left;"> Extra: If you get to a problem that has 2 sin x = 1 then you divide the other side be 2 to get sinx = 1/2 but if the problem says sin 2x = 1 then you have to do the inverse of sin and then solve for x like: sin 2x = 1 2x = sin^-1 (1) 2x = 90 x=45

__<span style="color: #bd32b5; font-family: 'Comic Sans MS',cursive; font-size: 140%;">Practice Problems __

1. cos x = 1 + sin x 2. cos x= 3 cos x -2 3. tan 2x= cot x

__<span style="color: #a32e95; font-family: 'Comic Sans MS',cursive; font-size: 160%;">Portfolio Problem __ <span style="color: #da1642; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 200%; text-align: left;">4sinx-3cosx=0 <span style="font-family: 'Comic Sans MS',cursive;"> My instructional video http://screencast-o-matic.com/watch/clf1fUkU2 <span style="color: #800080; display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 200%; text-align: left;">Sources []

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