Results 1 to 2 of 2

Thread: trigo show question

  1. #1
    Senior Member
    Joined
    Jan 2009
    Posts
    381

    trigo show question

    Show that $\displaystyle sin(\frac{\pi}{4}-\theta)=cos(\frac{\pi}{4}+\theta)$ and

    $\displaystyle sin(\frac{\pi}{4}+\theta)=cos(\frac{\pi}{4}-\theta)$

    i can do this part ..

    Show that $\displaystyle sin^4(\frac{\pi}{4}-\theta)+sin^4(\frac{\pi}{4}+\theta)=\frac{3}{4}-\frac{1}{4}cos4\theta $

    i cant do this part .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Trig proof

    Hello thereddevils
    Quote Originally Posted by thereddevils View Post
    Show that $\displaystyle sin(\frac{\pi}{4}-\theta)=cos(\frac{\pi}{4}+\theta)$ and

    $\displaystyle sin(\frac{\pi}{4}+\theta)=cos(\frac{\pi}{4}-\theta)$

    i can do this part ..

    Show that $\displaystyle sin^4(\frac{\pi}{4}-\theta)+sin^4(\frac{\pi}{4}+\theta)=\frac{3}{4}-\frac{1}{4}cos4\theta $

    i cant do this part .
    $\displaystyle \sin^4A = (\sin^2A)^2$

    $\displaystyle =\Big(\tfrac12(1-\cos2A)\Big)^2$

    $\displaystyle = \tfrac14(1-2\cos2A + \cos^22A)$

    $\displaystyle =\tfrac14(1 - 2\cos2A +\tfrac12(\cos4A +1))$

    $\displaystyle =\tfrac14(\tfrac32 - 2\cos2A +\tfrac12\cos4A)$

    If we now replace $\displaystyle A$ by $\displaystyle (\tfrac{\pi}{4}+\theta)$ and then by $\displaystyle (\tfrac{\pi}{4}-\theta)$, and add, we get:

    $\displaystyle \sin^4(\tfrac{\pi}{4}+\theta)+\sin^4 (\tfrac{\pi}{4}-\theta)=\tfrac14\Big(3 - 2[\cos(\tfrac{\pi}{2}+2\theta) + \cos(\tfrac{\pi}{2}-2\theta)]+\tfrac12[\cos(\pi+4\theta)+\cos(\pi-4\theta)]\Big)$

    Now use the formula for the sum of two cosines: $\displaystyle \cos A + \cos B = 2\cos\tfrac12(A+B)\cos\tfrac12(A-B)$

    $\displaystyle \sin^4(\tfrac{\pi}{4}+\theta)+\sin^4 (\tfrac{\pi}{4}-\theta)=\tfrac14(3 - 4\cos\tfrac{\pi}{2}\cos2\theta+\cos\pi\cos4\theta)$

    $\displaystyle = \tfrac34 -\tfrac14\cos4\theta$

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. need help for basic trigo question
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Jun 19th 2010, 07:47 AM
  2. trigo question
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Aug 10th 2009, 07:29 AM
  3. another trigo question
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: Dec 6th 2008, 11:27 PM
  4. Please help me with this GCSE trigo question
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: Oct 9th 2008, 02:17 PM
  5. trigo idnentitis question
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: Sep 9th 2008, 06:01 AM

Search Tags


/mathhelpforum @mathhelpforum