Results 1 to 6 of 6

Thread: Trigonometric equations of multiple angles

  1. #1
    Member
    Joined
    Mar 2017
    From
    Toronto
    Posts
    178
    Thanks
    1

    Question Trigonometric equations of multiple angles

    Hi,

    I hope someone can help. I'm trying to solve for the following:
    Trigonometric equations of multiple angles-19074879_1434819969907426_1147036990_o.jpg

    I don't think I did the last few steps correct. I'm not sure whether it was correct to apply the phase shift of 1 to the solutions before the horizontal compression of a 1/2. Any ideas?

    - Olivia
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,102
    Thanks
    2573
    Awards
    1

    Re: Trigonometric equations of multiple angles

    Quote Originally Posted by otownsend View Post
    Hi,
    I hope someone can help. I'm trying to solve for the following:
    Click image for larger version. 

Name:	19074879_1434819969907426_1147036990_o.jpg 
Views:	16 
Size:	92.0 KB 
ID:	37759
    I don't think I did the last few steps correct. I'm not sure whether it was correct to apply the phase shift of 1 to the solutions before the horizontal compression of a 1/2. Any ideas?
    Please learn to post the complete question. You seem to add that there are limitations on the domain?

    Just look at $\cos \left( {2x - \frac{\pi }{4}} \right) = \dfrac{{\sqrt 2 }}{2}$
    It is easily seen that $2x - \frac{\pi }{4}=\pm\dfrac{\pi}{4}+2k\pi$
    Note that $x=0$ is also a solution.
    Now you finish.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2017
    From
    Toronto
    Posts
    178
    Thanks
    1

    Re: Trigonometric equations of multiple angles

    Sorry. The purpose of this question is to solve for x within the domain of [0, 2pi)

    I will provide a specific example to explain where I am confused. So once you find the first solution for 2x-(pi/4), which is pi/4, I'm wondering what the steps are you take in order to manipulate that value to produce the value in terms of just x? Do you multiply pi/2 by 1/2 and then add pi/4 or do you add pi/4 first and then multiply by 1/2?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,102
    Thanks
    2573
    Awards
    1

    Re: Trigonometric equations of multiple angles

    Quote Originally Posted by otownsend View Post
    Sorry. The purpose of this question is to solve for x within the domain of [0, 2pi)

    I will provide a specific example to explain where I am confused. So once you find the first solution for 2x-(pi/4), which is pi/4, I'm wondering what the steps are you take in order to manipulate that value to produce the value in terms of just x? Do you multiply pi/2 by 1/2 and then add pi/4 or do you add pi/4 first and then multiply by 1/2?
    Zero is a solution. Recall that $\cos$ is an even function:
    $\cos \left( {2 \times 0 - \frac{\pi }{4}} \right) = \underbrace {\cos \left( {\frac{-\pi }{4}} \right)}_{even}= \cos \left( { \frac{\pi }{4}} \right) = \frac{{\sqrt 2 }}{2}$

    $\pi$ is also a solution: $2\times\pi-\frac{\pi}{4}=\frac{7\pi}{4}$ and $\cos\left(\frac{7\pi}{4}\right)=\frac{\sqrt2}{2}$ .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    15,883
    Thanks
    3531

    Re: Trigonometric equations of multiple angles

    $\cos \left(2x - \dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}$

    from the unit circle, cosine of an angle equals $\dfrac{\sqrt{2}}{2}$ at an angle of $\dfrac{\pi}{4}$ above and below the positive x-axis (quadrants I and IV)

    $0 \le x < 2\pi \implies -\dfrac{\pi}{4} \le 2x-\dfrac{\pi}{4} < \dfrac{15\pi}{4}$

    $2x-\dfrac{\pi}{4} = \bigg\{-\dfrac{\pi}{4}, \, \dfrac{\pi}{4},\, \dfrac{7\pi}{4}, \, \dfrac{9\pi}{4} \bigg\}$

    $2x = \bigg\{0, \, \dfrac{\pi}{2},\, 2\pi, \, \dfrac{5\pi}{2} \bigg\}$

    $x = \bigg\{0, \, \dfrac{\pi}{4},\, \pi, \, \dfrac{5\pi}{4} \bigg\}$
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2017
    From
    Toronto
    Posts
    178
    Thanks
    1

    Re: Trigonometric equations of multiple angles

    Oh okay I understand. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Trigonometric ratios of compound angles
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: May 31st 2017, 05:17 PM
  2. Trigonometric Ratios of Allied angles
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Jun 21st 2013, 04:52 AM
  3. Equations Involving Multiple Angles
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: May 3rd 2011, 04:37 AM
  4. Trigonometric Ratios and and Special Angles
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: Dec 2nd 2009, 03:22 AM
  5. Trigonometric Functions of Non-Acute Angles
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Sep 14th 2009, 05:20 PM

Search Tags


/mathhelpforum @mathhelpforum