Friday, December 16, 2016
5.3 Solving trig equations
Tips for solving/verifying trig equations?
I always start by deducing as many sides and angles as possible. That way you have plenty of options to solve the problem. It's good since some people tnd to forget some formula, there's always another path to the solution.
It also helps to write down the relationship between the parts of the triangle on a separate side. Sometimes I forget that I have enough information to conclude something and that gives me less time to complete the other questions.
If you want to verify your answer you can use different identities to see if they agree with the angles and sides you found in the triangle.
It also helps to write down the relationship between the parts of the triangle on a separate side. Sometimes I forget that I have enough information to conclude something and that gives me less time to complete the other questions.
If you want to verify your answer you can use different identities to see if they agree with the angles and sides you found in the triangle.
Here are some tips our math teacher gave us:
1. Change everything into sin and cos. It makes it a lot easier. Make sure you know the identities (you should have a list), such as sin x / cos x = tan x, or sin^2 x + cos^2 x = 1.
2. You may have to rearrange some of the identities. For example, to find what cos^2 x is in the 2nd formula above, subtract sin^2 x on both sides, so that you get cos^2 x = 1 - sin^2 x.
2. Look at miniproofs such as the ones above that you can substitute into the equation.
3. Form common denominators for all the parts of the equation. Or, cross-multiply.
4. FOIL (First, Inner, Outer, Last) if you see that you can.
1. Change everything into sin and cos. It makes it a lot easier. Make sure you know the identities (you should have a list), such as sin x / cos x = tan x, or sin^2 x + cos^2 x = 1.
2. You may have to rearrange some of the identities. For example, to find what cos^2 x is in the 2nd formula above, subtract sin^2 x on both sides, so that you get cos^2 x = 1 - sin^2 x.
2. Look at miniproofs such as the ones above that you can substitute into the equation.
3. Form common denominators for all the parts of the equation. Or, cross-multiply.
4. FOIL (First, Inner, Outer, Last) if you see that you can.
You must practice a lot to be good in any mathematics subject. Try solving sample exercises in books, this can really sharpen up your skills in trigonometry. Also, memorize those trigonometric identities and theorems.
5.2 Verifying trig identities
Trigonometric equations can be broken into two categories: identities and conditional equations. Identities are true for any angle, whereas conditional equations are true only for certain angles. Identities can be tested, checked, and created using knowledge of the eight fundamental identities. We already discussed these processes in Trigonometric Identities . The following sections are dedicated to explaining how to solve conditional equations.
Conditional Equations
When solving a conditional equation, a general rule applies: if there is one solution, then there are an infinite number of solutions. This strange truth results from the fact that the trigonometric functions are periodic, repeating every 360 degrees or 2Π radians. For example, the values of the trigonometric functions at 10 degrees are the same as they are at 370 degrees and 730 degrees. The form for any answer to a conditional equation is θ +2nΠ , where θ is one solution to the equation, and n is an integer. The shorter and more common way to express the solution to a conditional equation is to include all the solutions to the equation that fall within the bounds [0, 2Π) , and to omit the " +2nΠ " part of the solution. since it is assumed as part of the solution to any trigonometric equation. Because the set of values from 0 to 2Π contains the domain for all six trigonometric functions, if there is no solution to an equation between these bounds, then no solution exists.
Solutions for trigonometric equations follow no standard procedure, but there are a number of techniques that may help in finding a solution. These techniques are essentially the same as those used in solving algebraic equations, only now we are manipulating trigonometric functions: we can factor an expression to get different, more understandable expressions, we can multiply or divide through by a scalar, we can square or take the square root of both sides of an equation, etc. Also, using the eight fundamental identities, we can substitute certain functions for others, or break a functions down into two different ones, like expressing tangent using sine and cosine. In the problems below, we'll see just how helpful some of these techniques can be.
problem1
2 cos(x) - 1 = 0
2 cos(x) = 1
cos(x) =
x = ,
In this problem, we came up with two solutions in the range [0, 2Π) : x = , and x = . By adding 2nΠ to either of these solutions, where n is an integer, we could have an infinite number of solutions.
problem2
sin(x) = 2 cos2(x) - 1
sin(x) = 2(1 - sin2(x)) - 1
sin(x) = 1 - 2 sin2(x)
2 sin2(x) + sin(x) - 1 = 0
(sin(x) + 1)(2 sin(x) - 1) = 0
At this point, after factoring, we have two equations we need to deal with separately. First, we'll solve (sin(x) + 1) = 0 , and then we'll solve (2 sin(x) - 1) = 0
problem2a
sin(x) + 1 = 0
sin(x) = - 1
x =
2 sin(x) - 1 = 0
sin(x) =
x = ,
For the problem, then, we have three solutions: x = ,, . All of them check. Here is one more problem.
problem3
sec2(x) + cos2(x) = 2
1 + tan2(x) + 1 - sin2(x) = 2
tan2(x) = sin2(x)
= sin2(x)
cos2(x) = 1
cos(x) = ±1
x = 0, Π
Conditional Equations
When solving a conditional equation, a general rule applies: if there is one solution, then there are an infinite number of solutions. This strange truth results from the fact that the trigonometric functions are periodic, repeating every 360 degrees or 2Π radians. For example, the values of the trigonometric functions at 10 degrees are the same as they are at 370 degrees and 730 degrees. The form for any answer to a conditional equation is θ +2nΠ , where θ is one solution to the equation, and n is an integer. The shorter and more common way to express the solution to a conditional equation is to include all the solutions to the equation that fall within the bounds [0, 2Π) , and to omit the " +2nΠ " part of the solution. since it is assumed as part of the solution to any trigonometric equation. Because the set of values from 0 to 2Π contains the domain for all six trigonometric functions, if there is no solution to an equation between these bounds, then no solution exists.
Solutions for trigonometric equations follow no standard procedure, but there are a number of techniques that may help in finding a solution. These techniques are essentially the same as those used in solving algebraic equations, only now we are manipulating trigonometric functions: we can factor an expression to get different, more understandable expressions, we can multiply or divide through by a scalar, we can square or take the square root of both sides of an equation, etc. Also, using the eight fundamental identities, we can substitute certain functions for others, or break a functions down into two different ones, like expressing tangent using sine and cosine. In the problems below, we'll see just how helpful some of these techniques can be.
problem1
2 cos(x) - 1 = 0
2 cos(x) = 1
cos(x) =
x = ,
In this problem, we came up with two solutions in the range [0, 2Π) : x = , and x = . By adding 2nΠ to either of these solutions, where n is an integer, we could have an infinite number of solutions.
problem2
sin(x) = 2 cos2(x) - 1
sin(x) = 2(1 - sin2(x)) - 1
sin(x) = 1 - 2 sin2(x)
2 sin2(x) + sin(x) - 1 = 0
(sin(x) + 1)(2 sin(x) - 1) = 0
At this point, after factoring, we have two equations we need to deal with separately. First, we'll solve (sin(x) + 1) = 0 , and then we'll solve (2 sin(x) - 1) = 0
problem2a
sin(x) + 1 = 0
sin(x) = - 1
x =
2 sin(x) - 1 = 0
sin(x) =
x = ,
For the problem, then, we have three solutions: x = ,, . All of them check. Here is one more problem.
problem3
sec2(x) + cos2(x) = 2
1 + tan2(x) + 1 - sin2(x) = 2
tan2(x) = sin2(x)
= sin2(x)
cos2(x) = 1
cos(x) = ±1
x = 0, Π
Sunday, November 6, 2016
4.4- Graphing sin and cosine functions
something that you have to know is...
this formula is
so important if you want to understand the next step
How to define period, amplitude, phase shift and vertical shift of a periodic function?'
Period--- When does it start to repeat? For instance, for period of sin(x) is 2*pi because it repeats every 2*pi.
Amplitude--- How high does the graph go? In other words, the maximum value minus the minimum value. For instance, the amplitude of sin(x) is 2 because it goes from 1 to -1.
Phase Shift--- How far is the graph shifted over to the right? For example, the phase shift of sin(x) is 0 because it is not moved at all in the x direction.
Vertical Shift--- How far is the graph shifted up? For example, the vertical shift of sin(x) +1 is 1 because the graph is shifted up 1 unit of sin(x).
Amplitude--- How high does the graph go? In other words, the maximum value minus the minimum value. For instance, the amplitude of sin(x) is 2 because it goes from 1 to -1.
Phase Shift--- How far is the graph shifted over to the right? For example, the phase shift of sin(x) is 0 because it is not moved at all in the x direction.
Vertical Shift--- How far is the graph shifted up? For example, the vertical shift of sin(x) +1 is 1 because the graph is shifted up 1 unit of sin(x).
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