Its is nothing much to worry about this topic... It is just the same as we studied previously in First Chapter.. RECALL IT..!!
*sorry guys I couldn't able attach with any notes/examples :'(...
Hmmm... Seems to be to the end of chapter. Practice Makes Better..!!! Hope you guys practice alot..CHEEEERRRSsssss!!! :) :)
Dawn with Advanced Functions..!!
HI! Welcome..So this is my blog..I know it is boring though..! Just catch a glimpse on..! Who knows it might be a chance for us to Refresh the incredible brain..! HAVE A GOOD TOUR..!! :P
Saturday 15 October 2011
5.4 Solve Trigonometric Equation :
All about CALculations !!! :)
Example 1:
![$2sin\left( x\right) -1=0$](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tmfUW1pxMgbswjBtjVDAgGxCkck7BsFP4nQLtbuO3XmbbCokev5fZrA7YmlmmFNh1cizQADHkmet7tX0PCUmERbd_o3rmPU2NmlJoHOzAzA8wIl3kECUVT59Bp=s0-d)
There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.
![\begin{eqnarray*}&& \\
2\sin \left( x\right) -1 &=&0 \\
&& \\
\sin \left( x\right) &=&\displaystyle \frac{1}{2} \\
&& \\
&&
\end{eqnarray*}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tQrfBo6gD2EoS8vu5W5gFtQibSREONYeeeJKQ5Try9fnS0RAX9aoPUr7Zqju3xT_FLvAtdtbw9moMSjD2CoqTcN2tO_tKXAU_gyiAvh5H7p_WvOCMj8Ldr1a4=s0-d)
We know that the
therefore
The sine function is positive in quadrants I and II. The
is also equal to
Therefore, two of the solutions to the problem are
and ![$x=\displaystyle \frac{5\pi }{6}.
$](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uqqtyyGHFr2cjAau1oNGZJAL74cDN0RYDmW2WIqhjj6x9RfpuVQu0uQ1cjwhJgDLey5M-lTA--FefhdSPsRS6zh4FA9_NqMeXl-2nl2pCTWSH0Z0vhLdDIbp6X=s0-d)
Example 1:
There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.
We know that the
Example 2:
![$2\sin\left( 3x\right) -1=0$](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sZqc8vEq71sl3eAVskMlZoz4DFO8mEutx1vIIvR3R8ZiBK1V9MwRAGaAlQUBi7dujdUzihPmxYAZDHji1RltLpnoQnIwpRS8M2iqqhc6D0KLEDMD-csr_4S5iN=s0-d)
There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.
![\begin{eqnarray*}&& \\
2\sin \left( 3x\right) -1 &=&0 \\
&& \\
\sin \left( 3x\right) &=&\displaystyle \frac{1}{2} \\
&& \\
&&
\end{eqnarray*}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vx52YLhzLXlh6p3zG8Z-8nzPB6pYMivJTJ1c0fpew25vFiOrEdBA6zfe0WHxWF2_0hVi4bYtqI3zpXuNsG5JuOjK2Qc-NKRfIBHBYjTcpOERUH1ReJ2H90tyY=s0-d)
If we restriction the domain of the sine function to
, we can use the inverse sine function to solve for reference angle 3x and then x.
![\begin{eqnarray*}&& \\
\sin \left( 3x\right) &=&\displaystyle \frac{1}{2} \\
&...
...&=&\sin ^{-1}\left( \displaystyle \frac{1}{2
}\right) \\
&& \\
\end{eqnarray*}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vQmCstzzyjAw0OMZtyIvbbn5pzd7YeXHt1FZ8VshXzHiQFCuRiyWGo8J_Oytl5nPlkRTsxYwaXJ7guKESM4euKUicSZGya7lHjqrrq_mTrP-sMe6693Xes5pKT=s0-d)
![\begin{eqnarray*}3x &=&\sin ^{-1}\left( \displaystyle \frac{1}{2}\right) \\
&& ...
...\
x &\approx &0.174532925\ \mbox{ radians }\\
&& \\
&& \\
&&
\end{eqnarray*}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vwfwofWdRy5W9suw9zWRXtLfntRdYMmUtjlM6BAN_Dy20IhqEaRh4do0CBFDol9MtmitA1q0wEepHZBX9CUNmU2P3xC4-KR3kDVBtkKoJ66t622i2Ub5rt_3a8=s0-d)
We know that the
e function is positive in the first and the second quadrant. Therefore two of the solutions are the angle 3x that terminates in the first quadrant and the angle
that terminates in the second quadrant. We have already solved for 3x.
![\begin{eqnarray*}&& \\
\sin \left( \pi -3x\right) &=&\displaystyle \frac{1}{2} ...
...-3x &=&\sin ^{-1}\left( \displaystyle \frac{1}{2}\right) \\
&&
\end{eqnarray*}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vhU32fHpyw6hxIDNv7-npVfAG5P0qbiNrsX32YAh-cpSQ405LOZug4bs1LVEXXFjCXrNq_6DpCyv-Vlzzg8_IiP0gSE0Uvm6CbcZW5waNYQ7TiguWcYUssw6Hu=s0-d)
The solutions are
and ![$
x=\displaystyle \frac{\pi }{3}-\displaystyle \frac{1}{3}\sin ^{-1}\left( \displaystyle \frac{1}{2}\right) .\bigskip
\bigskip\bigskip $](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s949rjPPbHAmNFgySaUjW3pW_YMTQ-JQDoilxwBS-JvkUKTQqNclfM5wgBxR1RBC-2cwxrJkEM4Onr1ZmPMF-5vuKiVRjrF8xg8iLyRCzT58JYawRc3z9OG5uf9g=s0-d)
Example 3.
Solve the equation
for 0 ≤ θ < 2π.
Example 4.
Solve the equation 6 sin2θ − sin θ − 1 = 0 for 0 ≤ θ < 2π.
Solve the equation tan 2θ − cot 2θ = 0 for 0 ≤ θ < 2π.
![image](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tdIsQEguDs2kA7hpfO5vwNBoeitqUefcVr2Ux1-rp8lPPwIGtgCQCix2CNnPcszDbbtPdVuPsqF1KCLq0Vh-_6z0Y_rEMXcUGWOOctD-IDr6Ys6JOfzTJiCQ=s0-d)
tan2 2θ = 1
Since 0 ≤ θ < 2π , we need to consider values of 2θ such that 0 ≤ 2θ < 4π. Hence, solving the above equation, we have:
There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.
If we restriction the domain of the sine function to
We know that the
Example 3.
Solve the equation
Solving for cos θ gives us:
If
, then the reference angle is α = 1.3181.
So for
, we have θ in the first and 4th quadrants. So
θ = 1.3181 or 4.9651
For
, we have θ in the 2nd and 3rd quadrants. So
θ = 1.8235 or 4.4597
So θ= 1.3181, 1.8235, 4.4597 or 4.9651 radians.
Solve the equation 6 sin2θ − sin θ − 1 = 0 for 0 ≤ θ < 2π.
Factoring the LHS:
6 sin2 θ − sin θ − 1 = 0
(2 sin θ − 1)(3 sin θ + 1) = 0
So either
2 sin θ − 1 = 0Example 5.
sin θ = 1/2
θ will be in 1st and 2nd quadrants.
θ = 0.52360, 2.6180 (ie π/6, 5π/6)
Solve the equation tan 2θ − cot 2θ = 0 for 0 ≤ θ < 2π.
tan2 2θ = 1
tan 2θ = ± 1
Since 0 ≤ θ < 2π , we need to consider values of 2θ such that 0 ≤ 2θ < 4π. Hence, solving the above equation, we have:
So
* For More : http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx
5.3 Sinusoidal Functions of the Transformation Form..!!
The transformation of a sine or cosine function f(x) to g(x) has the general form g(x) = a f ( k(x - d) ) + c,
where :
where :
- a is the amplitude.
- d is the phase shift
- c is the vertical translation
The period of the transformed function is given by 2π
k
5.2 Graphs of Reciprocal Trigonometric Function
The graphs of y = cscx, y = sec x, and y = cot x are periodic. They are related to the graphs of the primary trigonometric functions as reciprocal graphs.
Cosecant
![Right triangle Cosecant function Right triangle Cosecant function](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u5DWcxYa2GiiMviEWyANjoFt2nlacPC_CfgP0vbLH0yayCSzW7wFt8gyKE6GrLFex5n-U9CPuZgk8Pg6dqUSvUquHKGKOx7IjSkwEwnaeq1LFaVU9FTXnQ6jAHA5lnYtcX2WzE7nZv0Q=s0-d)
![Graph of cosecant function Graph of cosecant function](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vskgIXdrIAkuzycg3mYAUOl5V5exnGgGtBYMngEQLXsfi4JiotqF-k7-T-uSmuWfaWm07HAnj_IMNibQdq4cqv3jKc82SY8zYhi74i7BwGz5V8oitQ7Ivcs8nYPOh-BIE-T4mSqOn-1jfoxZhRS3JZjbPCojaQ=s0-d)
Secant Graph.
Graph of Cotangent .
Below see cotangent function based on right angle triangle -
![Cotangent function Cotangent function](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_shrESU-riWg5llmgSMTVfHVtiTH5fdeqTCrTCu88zamRn8VOx2a6niE2rMtRzxKWp3hxWJ3uHaMSStG6uItj6RBzzx2jpY3k3IYLyi-xzcF5Y8jZAHt3zqeA7f24Gv_w=s0-d)
Graph for Cotangent function:
y = f (x) = cot x
![Graph of cotangent Graph of cotangent](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vVfeEJ1le-a5XegT7txUfZUJnqBSHUXEalDsoBEIgMrrWTzeo9tYvZNkA0ohAozbzqLae200Q9XkTHkRNH-ra2jz82BxT3HsMFxSsoYeIQHSyNqMlEHvLGl7rgxkpl=s0-d)
Graph of Cosecant
The Cosecant of an angleis the ratio of length of the hypotenuse and the opposite side.
csc (q) = hypotenuse / Opposite
The cosecant is the inverse of the sine and is given by the relation
csc q = 1 / sin q- Graph of cosecant function:
y = f (x) = csc (x)
Secant Graph.
Secant of an angleis the ratio of length of the hypotenuse and the opposite side.
Sec (q) = hypotenuse / opposite
The secant is the inverse of the sine and is given by the relation
sec q = 1 / cos q
sec q = 1 / cos q
- Co-function for secant
Relations between the secant and cosine functions are the co-function of the other.
Sec A = cos (90° - x)
Sec A = cos (90° - x)
y = f (x) = sec x
- An angleis the ratio of length of the adjacent side and the length of the opposite side is known as Cotangent.
Cot (q) = adjacent / opposite
- It is the inverse of the tangent and its relation is given as
Cot (q ) = 1 / tan q
- It is the ratio of cosine and sine functioncan be defined as
Cot (q) = cos q / sin q
y = f (x) = cot x
5.1 Graphs of Sine, Cosine, and Tangent Functions
KeyNotes :
![Tan](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tazmYcGDLah54L57Zi3ru6lg7QqEJ5EBMs6XwVJgQego5vcfdmqkWjiMk35qmQMgrWSJlgk8Z-rMCsnN5LyXxSlwqJxlDvNAYf9lCakxAI8UBNUWUe9P5JzUFQB8omJGU4Y4HEWnJOjbGLC5uk2JKZiyF4kOJj6TRtGqyJOEYdfD0VOfQuL4eNyzTaY7jGp4GTXEBi7MF8ii4kIrD5eJD-zeUkcreU=s0-d)
![Tan](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v9CGawQOmPGAa39bkkFL_U6-Rgn8L70688b0Y87fO7Z2iuiFDPWUiF9nJuf470aTNfB_ySLV8qQFfgrKc9b6wpJA8EjSGB47lWG9xbe3OYGr37e_TIc0lyEgZjsew7_Ktm0OBeZpuEiNTfLvPlnklGRM5SY4TMfSPYzal6MihrzWYnGcERqqfH-_YlE6CQjkkgcI48oEX1_P2hGY8_L7fUfHDWgnk=s0-d)
![Tan](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v_w9nnLN2VY-z67cX4vmAspta7QPWmNmgcgjDYpShGWgFYKUZEsgfPokBouI0E5LG7yPY_CZdwUoEcagUJplXEmoid20fAyQ2a3-oSt2v4-h14sAqHr1G0LuXSv1b4ICsg0j9gzu9_px7ThDP8kT2gHVbXrrGnMH2NlJg3g8mq9Wx7aGu4alZge0e5kAChAqZmNxE8hhvtoCubQXl6MCNmr4Cdq_s=s0-d)
- The graphs of y = sin x, y = cos x and y= tan x are periodic.
Sine and Cosine graphs.
- The graphs of y = sin x and y = cos x have an amplitude (finding by adding the max. and min. values then dividing by 2) of 1 and a period (a complete cycle) of 2π.
Tangent Graph.
The tan function
The tan function is found using:
It therefore follows that tan θ = 0, when sin θ = 0, and tan θ is undefined when cos θ = 0.
1. This graph is continuous, but is undefined when
2. The range of values for tan θ is unlimited.
3. It has a period of π.
- All of the three functions periodically repeat their values and the simplest way to learn this, is to make sure that you understand the general rules below which use 'n' to represent any integer (i.e. any whole number, both positive and negative).
Sunday 7 August 2011
Graphing Calculator..
Here is an useful link for those who have not familiar with graphing calculator! Technology made it easy! No need to turn over the pages and pages again..!
Please feel free to have a look on it..!
http://www.prenhall.com/divisions/esm/app/graphing/ti83/
Please feel free to have a look on it..!
http://www.prenhall.com/divisions/esm/app/graphing/ti83/
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